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Argument estimates of certain meromorphically multivalent functions associated with generalized hypergeometric function
Applied Mathematics and Computation 206 (2008) 772–780
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Argument estimates of certain meromorphically multivalent functions associated with generalized hypergeometric function M.K. Aouf Faculty of Science, Mansoura University, Mansoura 35516, Egypt
a r t i c l e
i n f o
a b s t r a c t The object of this paper is to obtain some argument properties of meromorphically multivalent functions associated with generalized hypergeometric function. We also derive the integral preserving properties in a sector. Ó 2008 Elsevier Inc. All rights reserved.
Keywords: Meromorphic Starlike Argument estimates Hypergeometric function
1. Introduction Let Rp denote the class of functions of the form
f ðzÞ ¼ zp þ
1 X
akp zkp
ðp 2 N ¼ f1; 2; . . .gÞ;
ð1:1Þ
k¼1
which are analytic and p-valent in the punctured disc U ¼ fz : z 2 C and 0 < jzj < 1g ¼ U n f0g. For functions f ðzÞ 2 Rp given by (1.1), and gðzÞ 2 Rp given by
gðzÞ ¼ zp þ
1 X
bkp zkp
ðp 2 NÞ;
ð1:2Þ
k¼1
we define the Hadamard product (or convolution) of f ðzÞ and gðzÞ by
ðf gÞðzÞ ¼ zp þ
1 X
akp bkp zkp :
ð1:3Þ
k¼1
For complex parameters a1 ; . . . ; aq and b1 ; . . . ; bs ðbj R Z 0 ¼ f0; 1; 2; . . .g; j ¼ 1; . . . ; sÞ, we now define the generalized hypergeometric function qFs(a1, . . . , aq; b1, . . . , bs; z) by
a
qFsð 1; . . . ;
aq ; b1 ; . . . ; bs ; zÞ ¼
1 X ða1 Þk . . . ðaq Þk zk : ðb1 Þk . . . ðbs Þk k! k¼0
ðq 6 s þ 1; q; s 2 N0 ¼ N [ f0g; z 2 UÞ;
ð1:4Þ
where ðhÞm is the Pochhammer symbol defined, in terms the Gamma function C, by
ðhÞm ¼
Cðh þ mÞ ¼ CðhÞ
1;
ðm ¼ 0; h 2 C n f0gÞ;
hðh þ 1Þ ðh þ m 1Þ; ðm 2 N; h 2 CÞ:
E-mail address: [email protected] 0096-3003/$ - see front matter Ó 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2008.09.046
ð1:5Þ
M.K. Aouf / Applied Mathematics and Computation 206 (2008) 772–780
773
Corresponding to a function hp ða1 ; . . . ; aq ; b1 ; . . . ; bs ; zÞ defined by
hp ða1 ; . . . ; aq ; b1 ; . . . ; bs ; zÞ ¼ zp q F s ða1 ; . . . ; aq ; b1 ; . . . ; bs ; zÞ; Liu and Srivastava [13] (see, for details, [6,7]) introduced a linear operator
Hp ða1 ; . . . ; aq ; b1 ; . . . ; bs Þ : Rp ! Rp ; which is defined by the following Hadamard product (or convolution)
Hp ða1 ; . . . ; aq ; b1 ; . . . ; bs Þf ðzÞ ¼ hp ða1 ; . . . ; aq ; b1 ; . . . ; bs ; zÞ f ðzÞ:
ð1:6Þ
We observe that, for a function f ðzÞ of the form (1.1), we have
Hp ða1 ; . . . ; aq ; b1 ; . . . ; bs Þf ðzÞ ¼
1 X ða1 Þk ðaq Þk akp kp : z : ðb1 Þk ðbs Þk k! k¼0
ð1:7Þ
If, for convenience, we write
Hp;q;s ða1 Þ ¼ Hp ða1 ; . . . ; aq ; b1 ; . . . ; bs Þ;
ð1:8Þ
then one can easily verify from the definition (1.6) that
zðHp;q;s ða1 Þf ðzÞÞ0 ¼ a1 Hp;q;s ða1 þ 1Þf ðzÞ ða1 pÞHp;q;s ða1 Þf ðzÞ:
ð1:9Þ
Some interesting subclasses of analytic functions, associated with the generalized hypergeometric function, were considered recently by (for example) Gangadharan et al. [9], Liu [11] and Aouf [2]. Let f ðzÞ and gðzÞ be analytic in U. Then we say that the function gðzÞ is subordinate to f ðzÞ if there exists an analytic function wðzÞ in U such that jwðzÞj < 1ðz 2 UÞ and gðzÞ ¼ f ðwðzÞÞ. For this subordination, the symbol gðzÞ f ðzÞ is used. In case f ðzÞ is univalent in U, the subordination gðzÞ f ðzÞ is equivalent to gð0Þ ¼ f ð0Þ and gðUÞ f ðUÞ. For a function f ðzÞ 2 Rp and m > 0 the integral operator F m;p ðf ÞðzÞ : Rp ! Rp is defined by
F m;p ðf ÞðzÞ ¼
m zmþp
Z
z
t mþp1 f ðtÞdt ¼
zp þ
0
1 X k¼1
m
mþk
! zpk f ðzÞ ¼ zp 2 F 1 ðm; 1; m þ 1; zÞ f ðzÞ ðm > 0; z 2 UÞ:
ð1:10Þ
It follows from (1.10) that
zðHp;q;s ðHp;q;s ða1 ÞF m;p ðf ÞðzÞÞÞ0 ¼ mHp;q;s ða1 Þf ðzÞ ðm þ pÞHp;q;s ða1 ÞF m;p ðf ÞðzÞ:
ð1:11Þ
The operator F m;p ðf ÞðzÞ was investigated by many authors (see for example [1,17,18]). We note that: (i) For q ¼ 2; s ¼ 1 and a2 ¼ 1, we obtain the linear operator
Hp;2;1 ða1 ; 1; b1 Þf ðzÞ ¼ ‘p ða1 ; b1 Þf ðzÞ ðf 2 Rp Þ; which was introduced and studied by Liu and Srivastava [12]. (ii) For any integer n > p and f ðzÞ 2 Rp , we have
Hp;2;1 ðn þ p; 1; 1Þf ðzÞ ¼ Dnþp1 f ðzÞ ¼
1 f ðzÞ; zp ð1 zÞnþp
where Dnþp1 f ðzÞ is the differential operator studied by Uralegaddi and Somanatha [17] and Aouf [1]. (iii) Hp;2;1 ðm; 1; m þ 1Þf ðzÞ ¼ F m;p ðf ÞðzÞðm > 0Þ. Let
Rp ½a1 ; A; B ¼ f 2 Rp :
zðHp;q;s ða1 Þf ðzÞÞ0 1 þ Az ; 1 6 B < A 6 1; z 2 U : p 1 þ Bz Hp;q;s ða1 Þf ðzÞ
ð1:12Þ
We note that: (i) For q ¼ 2; s ¼ 1; a1 ¼ b1 ¼ p; a2 ¼ 1; A ¼ 1 and B ¼ 1, we note that Rp ½p; 1; p; 1; 1 ¼ Rp is the well-known class of meromorphically starlike functions. h i (ii) q ¼ 2; s ¼ 1; a1 ¼ b1 ¼ p; a2 ¼ 1; A ¼ 1 2pa ; 0 6 a < p, and B ¼ 1, we note that Rp p; 1; p; 1 2pa ; 1 ¼ Rp ½a is the well-known class of meromorphically starlike functions of order a (see [3]). From (1.12) and by using the result of Silverman and Silvia [16], we observe that a function f ðzÞ is in Rp ½a1 ; A; B if and only if
zðHp;q;s ða1 Þf ðzÞÞ0 pð1 ABÞ pðA BÞ < þ Hp;q;s ða1 Þf ðzÞ 1 þ B2 1 B2
ð1 < B < A 6 1; z 2 UÞ:
ð1:13Þ
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M.K. Aouf / Applied Mathematics and Computation 206 (2008) 772–780
The object of the present paper is to give some argument properties of meromorphically functions belonging to Rp and the integral preserving properties in connection with the operator Hp;q;s ða1 Þ defined by (1.7). 2. Main result In order to show our main results, we need the following lemmas. Lemma 1 [8]. Let h be convex univalent in U with hð0Þ ¼ 1 and ReðbhðzÞ þ cÞ > 0ðb; c 2 CÞ. If q is analytic in U with qð0Þ ¼ 1, then
qðzÞ þ
zq0 ðzÞ hðzÞ ðz 2 UÞ bqðzÞ þ c
implies
qðzÞ hðzÞ ðz 2 UÞ: Lemma 2 [14]. Let h be convex univalent in U and kðzÞ be analytic in U Re kðzÞ P 0. If q is analytic in U and qð0Þ ¼ hð0Þ, then
qðzÞ þ kðzÞzq0 ðzÞ hðzÞ ðz 2 UÞ implies
qðzÞ hðzÞ ðz 2 UÞ: Lemma 3 [15]. Let q be analytic in U with qð0Þ ¼ 1 and qðzÞ–0 in U. Suppose that there exists a point z0 in U such that
j arg qðzÞj <
p 2
a for jzj < jz0 j
ð2:1Þ
and
j arg qðz0 Þj ¼
p 2
a ð0 < a 6 1Þ:
ð2:2Þ
Then we have
z0 q0 ðz0 Þ ¼ ika; qðz0 Þ
ð2:3Þ
where
1 1 p aþ when arg qðz0 Þ ¼ a; 2 a 2 1 1 p kP a aþ when arg qðz0 Þ ¼ 2 a 2 kP
ð2:4Þ ð2:5Þ
and 1
qðz0 Þa ¼ ia ða > 0Þ:
ð2:6Þ
At first, with the help of Lemma 1, we obtain the following theorem. Theorem 1. Let h be convex univalent in U with hð0Þ ¼ 1 and Re h be bounded in U. If f ðzÞ 2 Rp satisfies the condition
zðHp;q;s ða1 þ 1Þf ðzÞÞ0 hðzÞ ðz 2 UÞ; pHp;q;s ða1 þ 1Þf ðzÞ
zðHp;q;s ða1 Þf ðzÞÞ0 hðzÞ ðz 2 UÞ pHp;q;s ða1 Þf ðzÞ
then
for maxz2U Re hðzÞ < Re
a1 þp p
(provided Hp;q;s ða1 Þf ðzÞ–0 in U).
Proof. Let
qðzÞ ¼
zðHp;q;s ða1 Þf ðzÞÞ0 Hp;q;s ða1 Þf ðzÞ
ðz 2 UÞ:
M.K. Aouf / Applied Mathematics and Computation 206 (2008) 772–780
775
By using (1.9), we have
a1 þ p
qðzÞ
¼
p
a1 Hp;q;s ða1 þ 1Þf ðzÞ : pHp;q;s ða1 Þf ðzÞ
ð2:7Þ
Taking logarithmic derivatives in both sides of (2.7) and multiplying by z, we get
zq0 ðzÞ zðHp;q;s ða1 þ 1Þf ðzÞÞ0 þ qðzÞ ¼ hðzÞ ðz 2 UÞ: pqðzÞ þ ða1 þ pÞ pHp;q;s ða1 þ 1Þf ðzÞ
n o From Lemma 1, it follows that qðzÞ hðzÞ for Re hðzÞ þ a1pþp > 0ðz 2 UÞ, which means
zðHp;q;s ða1 Þf ðzÞÞ0 hðzÞ ðz 2 UÞ pHp;q;s ða1 Þf ðzÞ
for maxz2U Re hðzÞ < Re a1pþp . h
Taking q ¼ 2; s ¼ 1; a1 ¼ n þ pðn > pÞ; a2 ¼ b1 ¼ 1 and hðzÞ ¼ 1þz in Theorem 1, we obtain the result obtained by Aouf [1, 1z Theorem 1]. Using Lemmas 1 and 2 and Theorem 1, we now derive Theorem 2. Let f ðzÞ 2 Rp . Choose a1 2 R such that
a1 P
pðA BÞ ; 1þB
where 1 < B < A 6 1 and p 2 N. If
0 arg zðHp;q;s ða1 þ 1Þf ðzÞÞ c < p d ð0 6 c < p; 0 < d 6 1Þ 2 Hp;q;s ða1 þ 1ÞgðzÞ for some g 2 Rp ða1 þ 1; A; BÞ, then
0 arg zðHp;q;s ða1 Þf ðzÞÞ c < p a; 2 Hp;q;s ða1 ÞgðzÞ
where að0 < a 6 1Þ is the solution of the equation
d¼aþ
2
p
( tan1
when
tðA; BÞ ¼
2
p
)
a sin p2 ð1 tðA; BÞÞ a1 ð1BÞþpðABÞ þ a cos p2 ð1 tðA; BÞÞ 1B
"
pðA BÞ
1
sin
ð2:8Þ
#
ða1 þ pÞð1 B2 Þ pð1 ABÞ
:
ð2:9Þ
Proof. Let
qðzÞ ¼
1 zðHp;q;s ða1 Þf ðzÞÞ0 þc ðz 2 UÞ: pc Hp;q;s ða1 ÞgðzÞ
By using the identity (1.9), we have
ðp cÞzq0 ðzÞHp;q;s ða1 ÞgðzÞ þ ðp cÞqðzÞzðHp;q;s ða1 Þf ðzÞÞ0 þ czðHp;q;s ða1 ÞgðzÞÞ0 ¼ ða1 þ pÞzðHp;q;s ða1 Þf ðzÞÞ0 a1 zðHp;q;s ða1 þ 1Þf ðzÞÞ0 :
ð2:10Þ
Dividing (2.10) by Hp;q;s ða1 ÞgðzÞ and simplifying, we obtain
qðzÞ þ
zq0 ðzÞ 1 zðHp;q;s ða1 þ 1Þf ðzÞÞ0 ¼ þc ; rðzÞ þ a1 þ p pc Hp;q;s ða1 þ 1ÞgðzÞ
where
rðzÞ ¼
zðHp;q;s ða1 ÞgðzÞÞ0 : Hp;q;s ða1 ÞgðzÞ
Since gðzÞ 2 Rp ða1 þ 1; A; BÞ, from Theorem 1, we have
rðzÞ
1 þ Az ; 1 þ Bz
ð2:11Þ
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M.K. Aouf / Applied Mathematics and Computation 206 (2008) 772–780
using (1.13), we have p
rðzÞ þ a1 þ p ¼ qei2u ; where
a1 ð1 þ BÞ pðA BÞ 1þB
a1 ð1 BÞ þ pðA BÞ 1B
;
tðA; BÞ < u < tðA; BÞ;
where tðA; BÞ is given by (2.9).
Let h be a function which maps onto the angular domain w : j arg wj < p2 d with hð0Þ ¼ 1. Applying Lemma 2 for this h with kðzÞ ¼ rðzÞþ1 a1 þp, we see that Re qðzÞ > 0 in U and hence qðzÞ–0 in U. If there exists a point z0 2 U such that the conditions (2.1) and (2.2) are satisfied, then by Lemma 3, we have (2.3) under the restrictions (2.4)–(2.6). 1 At first, suppose that qðz0 Þa ¼ iaða > 0Þ. Then we obtain
arg
1 z0 ðHp;q;s ða1 þ 1Þf ðz0 ÞÞ0 z0 q0 ðz0 Þ p p þc ¼ a þ arg 1 þ iakðqei2u Þ1 ¼ arg qðz0 Þ þ pc rðz0 Þ þ a1 þ p Hp;q;s ða1 þ 1Þgðz0 Þ 2 ak sin p2 ð1 uÞ p ¼ a þ tan1 2 q þ ak cos p2 ð1 uÞ ! a sin p2 ð1 tðA; BÞÞ p p 1 P a þ tan ¼ d; a1 ð1BÞþpðABÞ 2 2 þ a cos p ð1 tðA; BÞÞ 1B
2
where d and tðA; BÞ are given by (2.8) and (2.9), respectively. This is a contradiction to the assumption of our theorem. 1 Next, suppose that pðz0 Þa ¼ iaða > 0Þ. Applying the same method as the above, we have
1 z0 ðHp;q;s ða1 þ 1Þf ðz0 ÞÞ0 p 6 a tan1 arg þc pc Hp;q;s ða1 þ 1Þgðz0 Þ 2
!
a sin p2 ð1 tðA; BÞÞ p ¼ d; a1 ð1BÞþpðABÞ p ð1 tðA; BÞÞ 2 þ a cos 1B 2
where d and tðA; BÞ are given by (2.8) and (2.9), respectively, which contradicts the assumption. Therefore we complete the proof of Theorem 2. h Taking A ¼ 1; B ¼ 0 and d ¼ 1 in Theorem 2, we have the following corollary. Corollary 1. Let f ðzÞ 2 Rp . If
zðHp;q;s ða1 þ 1Þf ðzÞÞ0 Re > c ð0 6 c < pÞ Hp;q;s ða1 þ 1ÞgðzÞ for some g 2 Rp satisfying the condition
zðHp;q;s ða1 þ 1ÞgðzÞÞ0 < p; þ p H ða þ 1ÞgðzÞ p;q;s 1 then
zðHp;q;s ða1 Þf ðzÞÞ0 Re > c ð0 6 c < pÞ: Hp;q;s ða1 ÞgðzÞ Remark 1. Taking q ¼ s þ 1; a1 ¼ b1 ¼ p; aj ¼ 1 ðj ¼ 2; 3; . . . ; s þ 1Þ and bj ¼ 1 ðj ¼ 2; . . . ; sÞ in Corollary 1 we obtain the result obtained by Lashin [10, Corollary 2.4]. Taking A ¼ 1; B ¼ 0 and gðzÞ ¼ z1p in Theorem 2, we have the following corollary. Corollary 2. Let f ðzÞ 2 Rp and a1 P p. If
j arg½zpþ1 ðHp;q;s ða1 þ 1Þf ðzÞÞ0 cj <
p 2
d ð0 6 c < p; 0 < c 6 1Þ;
then
j arg½zpþ1 ðHp;q;s ða1 Þf ðzÞÞ0 cj <
p 2
d:
Taking q ¼ s þ 1; a1 ¼ b1 ¼ p; aj ¼ 1 ðj ¼ 2; 3; . . . ; s þ 1Þ; bj ¼ 1 ðj ¼ 2; 3; . . . ; sÞ and d ¼ 1 in Corollary 2, we have the following corollary. Corollary 3. Let f ðzÞ 2 Rp . If 00
Refzpþ1 ½zf ðzÞ þ ð2p þ 1Þf 0 ðzÞg > c ð0 6 c < pÞ;
M.K. Aouf / Applied Mathematics and Computation 206 (2008) 772–780
777
then
Refzpþ1 f 0 ðzÞg > c: Remark 2. Taking q ¼ 2; s ¼ 1; a1 ¼ m; a2 ¼ 1 and b1 ¼ m þ 1; m > 1, in Theorem 2, we obtain the result obtained by Lashin [10, Corollary 2.3]. Remark 3 (i) Putting q ¼ 2; s ¼ 1; a1 ¼ n þ pðn > pÞ; a2 ¼ 1 and b1 ¼ 1 in Theorem 2, we obtain the result obtained by Cho and Owa [5, Theorem 2.1]. (ii) Putting p ¼ 1; q ¼ 2; s ¼ 1; a1 ¼ n þ 1ðn > 1Þ; a2 ¼ 1 and b1 ¼ 1 in Theorem 2, we obtain the result obtained by Cho [4, Theorem 2.1]. (iii) Putting q ¼ 2; s ¼ 1; a1 ¼ a > 0; a2 ¼ 1 and b1 ¼ c > 0 in Theorem 2, we obtain the result obtained by Lashin [10, Theorem 2.2]. By the same techniques as in the proof of Theorem 2, we obtain Theorem 3. Let f ðzÞ 2 Rp . Choose a1 2 R such that
a1 P
pðA BÞ ; 1þB
where 1 < B < A 6 1 and p 2 N. If
0 arg zðHp;q;s ða1 þ 1Þf ðzÞÞ þ c < p d ðc > p; 0 < d 6 1Þ 2 Hp;q;s ða1 þ 1ÞgðzÞ for some g 2 Rp ½a1 þ 1; A; B, then
0 arg zðHp;q;s ða1 Þf ðzÞÞ þ c < p a; 2 Hp;q;s ða1 ÞgðzÞ
where að0 < a 6 1Þ is the solution of the equation given by (2.8). Theorem 4. Let h be convex univalent in U with hð0Þ ¼ 1 and Re h be bounded in U. Let F m;p ðf ÞðzÞ be the integral operator defined by (1.10). If f 2 Rp satisfies the condition
zðHp;q;s ða1 Þf ðzÞÞ0 hðzÞ ðz 2 UÞ; pHp;q;s ða1 Þf ðzÞ
zðHp;q;s ða1 ÞF m;p ðf ÞðzÞÞ0 hðzÞ ðz 2 UÞ pHp;q;s ða1 ÞF m;p ðf ÞðzÞ
then
for maxz2U Re hðzÞ < mþp (provided Hp;q;s F m;p ðf ÞðzÞ–0 in U). p Proof. Let
pðzÞ ¼
zðHp;q;s ða1 ÞF m;p ðf ÞðzÞÞ0 pHp;q;s ða1 ÞF m;p ðf ÞðzÞ
ðz 2 UÞ:
Then, by using (1.11), we have
qðzÞ ðm þ pÞ ¼ c
Hp;q;s ða1 ÞF m;p ðf ÞðzÞ : Hp;q;s ða1 ÞF m;p ðf ÞðzÞ
Taking logarithmic derivatives in both sides of (2.12) and multiplying by z, we get
zq0 ðzÞ zðHp;q;s ða1 Þf ðzÞÞ0 þ qðzÞ ¼ hðzÞ ðz 2 UÞ: pqðzÞ þ ðm þ pÞ pHp;q;s ða1 Þf ðzÞ Therefore, by using Lemma 1, we have
zðHp;q;s ða1 ÞF m;p ðf ÞðzÞÞ0 hðzÞ ðz 2 UÞ pHp;q;s ða1 ÞF m;p ðf ÞðzÞ
for maxz2U Re hðzÞ < mþp (provided Hp;q;s ða1 ÞF m;p ðf ÞðzÞ–0 in U). h p
ð2:12Þ
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M.K. Aouf / Applied Mathematics and Computation 206 (2008) 772–780
1þz Remark 4. Taking q ¼ 2; s ¼ 1; a1 ¼ n þ pðn > pÞ; a2 ¼ b1 ¼ 1 and hðzÞ ¼ 1z in Theorem 4, we obtain the result obtained by Aouf [1, Theorem 3].
Theorem 5. Let f ðzÞ 2 Rp and choose a positive number
mP
m such that
1þA p; 1þB
where 1 < B < A 6 1 and p 2 N. If
0 arg zðHp;q;s ða1 Þf ðzÞÞ c < p d ð0 6 c < p; 0 < d 6 1Þ 2 Hp;q;s ða1 ÞgðzÞ for some g 2 Rp ½a1 ; A; B, then
0 arg zðHp;q;s ða1 ÞF m;p ðf ÞðzÞÞ c < p a; 2 Hp;q;s ða1 ÞGm;p ðgÞðzÞ where F m;p ðf ÞðzÞ is the integral operator given by (1.10),
Gm;p ðgÞðzÞ ¼
Z
m zmþp
z
tmþp1 gðtÞdt
ðm > 0Þ
ð2:13Þ
0
and að0 < a 6 1Þ is the solution of the equation
d¼aþ
2
p
(
)
a sin p2 ð1 tðA; B; mÞÞ ; ðmþpÞð1BÞþpðABÞ þ a cos p2 ð1 tðA; B; mÞÞ 1B
1
tan
ð2:14Þ
where
tðA; B; mÞ ¼
2
p
" 1
sin
pðA BÞ ðm þ pÞð1 B2 Þ pð1 ABÞ
# ð2:15Þ
:
Proof. Let
qðzÞ ¼
1 zðHp;q;s ða1 ÞF m;p ðf ÞðzÞÞ0 þc ðz 2 UÞ: p c Hp;q;s ða1 ÞGm;p ðgÞðzÞ
Since g 2 Rp ½a1 ; A; B, from Theorem 4, Gm;p ðgÞðzÞ 2 Rp ½a1 ; A; B. Using (1.11), we have
ðp cÞqðzÞHp;q;s ða1 ÞGm;p ðgÞðzÞ ðm þ pÞHp;q;s ða1 ÞF m;p ðf ÞðzÞ ¼ mHp;q;s ða1 Þf ðzÞ cHp;q;s ða1 ÞGm;p ðgÞðzÞ: Then, by a simple calculation, we have
ðp cÞfzq0 ðzÞ þ qðzÞ½rðzÞ þ m þ pg þ c½rðzÞ þ m þ p ¼
zðHp;q;s ða1 Þf ðzÞÞ0 ; Hp;q;s ða1 ÞGm;p ðgÞðzÞ
where
rðzÞ ¼
zðHp;q;s ða1 ÞGm;p ðgÞðzÞÞ0 : Hp;q;s ða1 ÞGm;p ðgÞðzÞ
Hence, we have
qðzÞ þ
zq0 ðzÞ 1 zðHp;q;s ða1 Þf ðzÞÞ0 ¼ þc : rðzÞ þ m þ p pc Hp;q;s ða1 ÞgðzÞ
The remaining part of the proof is similar to that of Theorem 2 and so we omit it.
h
Putting q ¼ s þ 1; a1 ¼ b1 ¼ p; aj ¼ 1 ðj ¼ 2; 3; . . . ; s þ 1Þ; bj ¼ 1 ðj ¼ 2; 3; . . . ; sÞ; A ¼ 1; B ¼ 0 and d ¼ 1 in Theorem 5, we obtain the following result. Corollary 4. Let
m > 0 and f ðzÞ 2 Rp . If
0 zf ðzÞ Re > c ð0 6 c < pÞ gðzÞ for some gðzÞ 2 Rp satisfying the condition
M.K. Aouf / Applied Mathematics and Computation 206 (2008) 772–780
779
0 zg ðzÞ < p; þ p gðzÞ then
Re
( 0 ) zF m;p ðf ÞðzÞ > c ð0 6 c < pÞ; Gm;p ðgÞðzÞ
where F m;p ðf ÞðzÞ and Gm;p ðgÞðzÞ are given by (1.10) and (2.13), respectively. Taking q ¼ s þ 1; a1 ¼ b1 ¼ p; aj ¼ 1 ðj ¼ 2; 3; . . . ; s þ 1Þ; bj ¼ 1 ðj ¼ 2; 3; . . . ; sÞ; B ! A and gðzÞ ¼ z1p in Theorem 5, we obtain the following corollary. Corollary 5. Let
j argðz
m > 0 and f ðzÞ 2 Rp . If
p
pþ1 0
f ðzÞ cÞj <
2
d ð0 6 c < p; 0 < d 6 1Þ
then
j argðzpþ1 F 0m;p ðf ÞðzÞ cÞj <
p 2
a;
where F m;p ðf ÞðzÞ is the integral operator given by (1.10) and að0 < a 6 1Þ is the solution of the equation
d¼aþ
2
p
tan1
a
mþp
:
By using the same method as in proving Theorem 5, we have Theorem 6. Let f ðzÞ 2 Rp and choose a positive number
m such that
1þA mP p; 1þB where 1 < B < A 6 1 and p 2 N. If
0 arg zðHp;q;s ða1 Þf ðzÞÞ þ c < p d ðc > p; 0 < d 6 1Þ 2 Hp;q;s ða1 ÞgðzÞ for some g 2 Rp ½a1 ; A; B, then
0 arg zðHp;q;s ða1 ÞF m;p ðf ÞðzÞÞ þ c < p a; 2 Hp;q;s ða1 ÞGm;p ðgÞðzÞ
where F m;p ðf ÞðzÞ and Gm;p ðgÞðzÞ are given by (1.10) and (2.13), respectively, and að0 < a 6 1Þ is the solution of the equation given by (2.14). Finally, we derive Theorem 7. Let f ðzÞ 2 Rp . Choose a1 2 R such that
a1 P
pðA BÞ ; 1þB
where 1 < B < A 6 1 and p 2 N. If
0 arg zðHp;q;s ða1 Þf ðzÞÞ c < p d ð0 6 c < p; 0 < d 6 1Þ 2 Hp;q;s ða1 ÞgðzÞ for some g 2 Rp ½a1 ; A; B, then
0 arg zðHp;q;s ða1 þ 1ÞF m;p ðf ÞðzÞÞ c < p d; 2 Hp;q;s ða1 þ 1ÞGm;p ðgÞðzÞ
where F m;p ðf ÞðzÞ and Gm;p ðgÞðzÞ are given by (1.10) and (2.13), respectively, with Proof. From (1.9) and (1.11) with
m ¼ a1 , we have Hp;q;s ða1 Þf ðzÞ ¼ Hp;q;s ða1 þ 1ÞF m;p ðf ÞðzÞ. Therefore
zðHp;q;s ða1 Þf ðzÞÞ0 zðHp;q;s ða1 þ 1ÞF m;p ðf ÞðzÞÞ0 ¼ Hp;q;s ða1 ÞgðzÞ Hp;q;s ða1 þ 1ÞGm;p ðgÞðzÞ and the result follows. h
m ¼ a1 .
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M.K. Aouf / Applied Mathematics and Computation 206 (2008) 772–780
References [1] M.K. Aouf, New criteria for multivalent meromorphic starlike function of order alpha, Proc. Jpn. Acad. Ser. A 69 (3) (1993) 66–70. [2] M.K. Aouf, Certain subclasses of meromorphically multivalent functions associated with generalized hypergeometric function, Comput. Math. Appl. 55 (3) (2008) 494–509. [3] M.K. Aouf, Certain classes of meromorphic multivalent functions with positive coefficients, Math. Comput. Model. 47 (3-4) (2008) 328–340. [4] N.E. Cho, Argument estimates of certain meromorphic functions, Comm. Korean Math. Soc. 15 (2) (2000) 263–274. [5] N.E. Cho, S. Owa, Argument estimates of meromorphically multivalent functions, J. Inequal. Appl. 5 (2000) 49–432. [6] J. Dziok, H.M. Srivastava, Classes of analytic functions associated with the generalized hypergeometric function, Appl. Math. Comput. 103 (1999) 1–13. [7] J. Dziok, H.M. Srivastava, Certain subclasses of analytic functions associated with the generalized hypergeometric function, Integral Transforms Spec. Funct. 14 (2003) 7–18. [8] P. Enigenburg, S.S. Miller, P.T. Mocanu, M.O. Reade, On a Briot-Bouquet differential subordination, General Inequal. 3 (1983) 339–348. [9] A. Gangadharan, T.N. Shanmugan, H.M. Srivastava, Generalized hypergeometric functions with k-uniformly convex functions, Comput. Math. Appl. 44 (12) (2002) 1515–1526. [10] A.Y. Lashin, Argument estimates of certain meromorphically p-valent functions, Soochow J. Math. 33 (4) (2007) 803–812. [11] J.-L. Liu, Strongly starlike functions associated with the Dziok–Srivastava operator, Tamkang J. Math. 35 (1) (2004) 37–42. [12] J.-L. Liu, H.M. Srivastava, A linear operator and associated families of meromorphically multivalent functions, J. Math. Anal. Appl. 259 (2000) 566–581. [13] J.-L. Liu, H.M. Srivastava, Classes of meromorphically multivalent functions associated with the generalized hypergeometric function, Math. Comput. Model. 39 (2004) 21–34. [14] S.S. Miller, P.T. Mocanu, Differential subordinations and univalent functions, Michigan Math. J. 28 (1981) 157–171. [15] M. Nunokawa, On the order of strongly starlikeness of strongly convex functions, Proc. Jpn. Acad. Ser. A – Math. Sci. 69 (1993) 234–237. [16] H. Silverman, E.M. Silvia, Subclasses of starlike functions subordinate to convex functions, Canad. J. Math. 37 (1985) 48–61. [17] B.A. Uralegaddi, C. Somanatha, Certain classes of meromorphic multivalent functions, Tamkang J. Math. 23 (1992) 223–231. [18] D. Yang, On a class of meromorphic starlike multivalent functions, Bull. Inst. Math. Acad. Sinica 24 (1996) 151–157.
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Argument estimates of certain meromorphically multivalent functions associated with generalized hypergeometric function M.K. Aouf Faculty of Science, Mansoura University, Mansoura 35516, Egypt
a r t i c l e
i n f o
a b s t r a c t The object of this paper is to obtain some argument properties of meromorphically multivalent functions associated with generalized hypergeometric function. We also derive the integral preserving properties in a sector. Ó 2008 Elsevier Inc. All rights reserved.
Keywords: Meromorphic Starlike Argument estimates Hypergeometric function
1. Introduction Let Rp denote the class of functions of the form
f ðzÞ ¼ zp þ
1 X
akp zkp
ðp 2 N ¼ f1; 2; . . .gÞ;
ð1:1Þ
k¼1
which are analytic and p-valent in the punctured disc U ¼ fz : z 2 C and 0 < jzj < 1g ¼ U n f0g. For functions f ðzÞ 2 Rp given by (1.1), and gðzÞ 2 Rp given by
gðzÞ ¼ zp þ
1 X
bkp zkp
ðp 2 NÞ;
ð1:2Þ
k¼1
we define the Hadamard product (or convolution) of f ðzÞ and gðzÞ by
ðf gÞðzÞ ¼ zp þ
1 X
akp bkp zkp :
ð1:3Þ
k¼1
For complex parameters a1 ; . . . ; aq and b1 ; . . . ; bs ðbj R Z 0 ¼ f0; 1; 2; . . .g; j ¼ 1; . . . ; sÞ, we now define the generalized hypergeometric function qFs(a1, . . . , aq; b1, . . . , bs; z) by
a
qFsð 1; . . . ;
aq ; b1 ; . . . ; bs ; zÞ ¼
1 X ða1 Þk . . . ðaq Þk zk : ðb1 Þk . . . ðbs Þk k! k¼0
ðq 6 s þ 1; q; s 2 N0 ¼ N [ f0g; z 2 UÞ;
ð1:4Þ
where ðhÞm is the Pochhammer symbol defined, in terms the Gamma function C, by
ðhÞm ¼
Cðh þ mÞ ¼ CðhÞ
1;
ðm ¼ 0; h 2 C n f0gÞ;
hðh þ 1Þ ðh þ m 1Þ; ðm 2 N; h 2 CÞ:
E-mail address: [email protected] 0096-3003/$ - see front matter Ó 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2008.09.046
ð1:5Þ
M.K. Aouf / Applied Mathematics and Computation 206 (2008) 772–780
773
Corresponding to a function hp ða1 ; . . . ; aq ; b1 ; . . . ; bs ; zÞ defined by
hp ða1 ; . . . ; aq ; b1 ; . . . ; bs ; zÞ ¼ zp q F s ða1 ; . . . ; aq ; b1 ; . . . ; bs ; zÞ; Liu and Srivastava [13] (see, for details, [6,7]) introduced a linear operator
Hp ða1 ; . . . ; aq ; b1 ; . . . ; bs Þ : Rp ! Rp ; which is defined by the following Hadamard product (or convolution)
Hp ða1 ; . . . ; aq ; b1 ; . . . ; bs Þf ðzÞ ¼ hp ða1 ; . . . ; aq ; b1 ; . . . ; bs ; zÞ f ðzÞ:
ð1:6Þ
We observe that, for a function f ðzÞ of the form (1.1), we have
Hp ða1 ; . . . ; aq ; b1 ; . . . ; bs Þf ðzÞ ¼
1 X ða1 Þk ðaq Þk akp kp : z : ðb1 Þk ðbs Þk k! k¼0
ð1:7Þ
If, for convenience, we write
Hp;q;s ða1 Þ ¼ Hp ða1 ; . . . ; aq ; b1 ; . . . ; bs Þ;
ð1:8Þ
then one can easily verify from the definition (1.6) that
zðHp;q;s ða1 Þf ðzÞÞ0 ¼ a1 Hp;q;s ða1 þ 1Þf ðzÞ ða1 pÞHp;q;s ða1 Þf ðzÞ:
ð1:9Þ
Some interesting subclasses of analytic functions, associated with the generalized hypergeometric function, were considered recently by (for example) Gangadharan et al. [9], Liu [11] and Aouf [2]. Let f ðzÞ and gðzÞ be analytic in U. Then we say that the function gðzÞ is subordinate to f ðzÞ if there exists an analytic function wðzÞ in U such that jwðzÞj < 1ðz 2 UÞ and gðzÞ ¼ f ðwðzÞÞ. For this subordination, the symbol gðzÞ f ðzÞ is used. In case f ðzÞ is univalent in U, the subordination gðzÞ f ðzÞ is equivalent to gð0Þ ¼ f ð0Þ and gðUÞ f ðUÞ. For a function f ðzÞ 2 Rp and m > 0 the integral operator F m;p ðf ÞðzÞ : Rp ! Rp is defined by
F m;p ðf ÞðzÞ ¼
m zmþp
Z
z
t mþp1 f ðtÞdt ¼
zp þ
0
1 X k¼1
m
mþk
! zpk f ðzÞ ¼ zp 2 F 1 ðm; 1; m þ 1; zÞ f ðzÞ ðm > 0; z 2 UÞ:
ð1:10Þ
It follows from (1.10) that
zðHp;q;s ðHp;q;s ða1 ÞF m;p ðf ÞðzÞÞÞ0 ¼ mHp;q;s ða1 Þf ðzÞ ðm þ pÞHp;q;s ða1 ÞF m;p ðf ÞðzÞ:
ð1:11Þ
The operator F m;p ðf ÞðzÞ was investigated by many authors (see for example [1,17,18]). We note that: (i) For q ¼ 2; s ¼ 1 and a2 ¼ 1, we obtain the linear operator
Hp;2;1 ða1 ; 1; b1 Þf ðzÞ ¼ ‘p ða1 ; b1 Þf ðzÞ ðf 2 Rp Þ; which was introduced and studied by Liu and Srivastava [12]. (ii) For any integer n > p and f ðzÞ 2 Rp , we have
Hp;2;1 ðn þ p; 1; 1Þf ðzÞ ¼ Dnþp1 f ðzÞ ¼
1 f ðzÞ; zp ð1 zÞnþp
where Dnþp1 f ðzÞ is the differential operator studied by Uralegaddi and Somanatha [17] and Aouf [1]. (iii) Hp;2;1 ðm; 1; m þ 1Þf ðzÞ ¼ F m;p ðf ÞðzÞðm > 0Þ. Let
Rp ½a1 ; A; B ¼ f 2 Rp :
zðHp;q;s ða1 Þf ðzÞÞ0 1 þ Az ; 1 6 B < A 6 1; z 2 U : p 1 þ Bz Hp;q;s ða1 Þf ðzÞ
ð1:12Þ
We note that: (i) For q ¼ 2; s ¼ 1; a1 ¼ b1 ¼ p; a2 ¼ 1; A ¼ 1 and B ¼ 1, we note that Rp ½p; 1; p; 1; 1 ¼ Rp is the well-known class of meromorphically starlike functions. h i (ii) q ¼ 2; s ¼ 1; a1 ¼ b1 ¼ p; a2 ¼ 1; A ¼ 1 2pa ; 0 6 a < p, and B ¼ 1, we note that Rp p; 1; p; 1 2pa ; 1 ¼ Rp ½a is the well-known class of meromorphically starlike functions of order a (see [3]). From (1.12) and by using the result of Silverman and Silvia [16], we observe that a function f ðzÞ is in Rp ½a1 ; A; B if and only if
zðHp;q;s ða1 Þf ðzÞÞ0 pð1 ABÞ pðA BÞ < þ Hp;q;s ða1 Þf ðzÞ 1 þ B2 1 B2
ð1 < B < A 6 1; z 2 UÞ:
ð1:13Þ
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M.K. Aouf / Applied Mathematics and Computation 206 (2008) 772–780
The object of the present paper is to give some argument properties of meromorphically functions belonging to Rp and the integral preserving properties in connection with the operator Hp;q;s ða1 Þ defined by (1.7). 2. Main result In order to show our main results, we need the following lemmas. Lemma 1 [8]. Let h be convex univalent in U with hð0Þ ¼ 1 and ReðbhðzÞ þ cÞ > 0ðb; c 2 CÞ. If q is analytic in U with qð0Þ ¼ 1, then
qðzÞ þ
zq0 ðzÞ hðzÞ ðz 2 UÞ bqðzÞ þ c
implies
qðzÞ hðzÞ ðz 2 UÞ: Lemma 2 [14]. Let h be convex univalent in U and kðzÞ be analytic in U Re kðzÞ P 0. If q is analytic in U and qð0Þ ¼ hð0Þ, then
qðzÞ þ kðzÞzq0 ðzÞ hðzÞ ðz 2 UÞ implies
qðzÞ hðzÞ ðz 2 UÞ: Lemma 3 [15]. Let q be analytic in U with qð0Þ ¼ 1 and qðzÞ–0 in U. Suppose that there exists a point z0 in U such that
j arg qðzÞj <
p 2
a for jzj < jz0 j
ð2:1Þ
and
j arg qðz0 Þj ¼
p 2
a ð0 < a 6 1Þ:
ð2:2Þ
Then we have
z0 q0 ðz0 Þ ¼ ika; qðz0 Þ
ð2:3Þ
where
1 1 p aþ when arg qðz0 Þ ¼ a; 2 a 2 1 1 p kP a aþ when arg qðz0 Þ ¼ 2 a 2 kP
ð2:4Þ ð2:5Þ
and 1
qðz0 Þa ¼ ia ða > 0Þ:
ð2:6Þ
At first, with the help of Lemma 1, we obtain the following theorem. Theorem 1. Let h be convex univalent in U with hð0Þ ¼ 1 and Re h be bounded in U. If f ðzÞ 2 Rp satisfies the condition
zðHp;q;s ða1 þ 1Þf ðzÞÞ0 hðzÞ ðz 2 UÞ; pHp;q;s ða1 þ 1Þf ðzÞ
zðHp;q;s ða1 Þf ðzÞÞ0 hðzÞ ðz 2 UÞ pHp;q;s ða1 Þf ðzÞ
then
for maxz2U Re hðzÞ < Re
a1 þp p
(provided Hp;q;s ða1 Þf ðzÞ–0 in U).
Proof. Let
qðzÞ ¼
zðHp;q;s ða1 Þf ðzÞÞ0 Hp;q;s ða1 Þf ðzÞ
ðz 2 UÞ:
M.K. Aouf / Applied Mathematics and Computation 206 (2008) 772–780
775
By using (1.9), we have
a1 þ p
qðzÞ
¼
p
a1 Hp;q;s ða1 þ 1Þf ðzÞ : pHp;q;s ða1 Þf ðzÞ
ð2:7Þ
Taking logarithmic derivatives in both sides of (2.7) and multiplying by z, we get
zq0 ðzÞ zðHp;q;s ða1 þ 1Þf ðzÞÞ0 þ qðzÞ ¼ hðzÞ ðz 2 UÞ: pqðzÞ þ ða1 þ pÞ pHp;q;s ða1 þ 1Þf ðzÞ
n o From Lemma 1, it follows that qðzÞ hðzÞ for Re hðzÞ þ a1pþp > 0ðz 2 UÞ, which means
zðHp;q;s ða1 Þf ðzÞÞ0 hðzÞ ðz 2 UÞ pHp;q;s ða1 Þf ðzÞ
for maxz2U Re hðzÞ < Re a1pþp . h
Taking q ¼ 2; s ¼ 1; a1 ¼ n þ pðn > pÞ; a2 ¼ b1 ¼ 1 and hðzÞ ¼ 1þz in Theorem 1, we obtain the result obtained by Aouf [1, 1z Theorem 1]. Using Lemmas 1 and 2 and Theorem 1, we now derive Theorem 2. Let f ðzÞ 2 Rp . Choose a1 2 R such that
a1 P
pðA BÞ ; 1þB
where 1 < B < A 6 1 and p 2 N. If
0 arg zðHp;q;s ða1 þ 1Þf ðzÞÞ c < p d ð0 6 c < p; 0 < d 6 1Þ 2 Hp;q;s ða1 þ 1ÞgðzÞ for some g 2 Rp ða1 þ 1; A; BÞ, then
0 arg zðHp;q;s ða1 Þf ðzÞÞ c < p a; 2 Hp;q;s ða1 ÞgðzÞ
where að0 < a 6 1Þ is the solution of the equation
d¼aþ
2
p
( tan1
when
tðA; BÞ ¼
2
p
)
a sin p2 ð1 tðA; BÞÞ a1 ð1BÞþpðABÞ þ a cos p2 ð1 tðA; BÞÞ 1B
"
pðA BÞ
1
sin
ð2:8Þ
#
ða1 þ pÞð1 B2 Þ pð1 ABÞ
:
ð2:9Þ
Proof. Let
qðzÞ ¼
1 zðHp;q;s ða1 Þf ðzÞÞ0 þc ðz 2 UÞ: pc Hp;q;s ða1 ÞgðzÞ
By using the identity (1.9), we have
ðp cÞzq0 ðzÞHp;q;s ða1 ÞgðzÞ þ ðp cÞqðzÞzðHp;q;s ða1 Þf ðzÞÞ0 þ czðHp;q;s ða1 ÞgðzÞÞ0 ¼ ða1 þ pÞzðHp;q;s ða1 Þf ðzÞÞ0 a1 zðHp;q;s ða1 þ 1Þf ðzÞÞ0 :
ð2:10Þ
Dividing (2.10) by Hp;q;s ða1 ÞgðzÞ and simplifying, we obtain
qðzÞ þ
zq0 ðzÞ 1 zðHp;q;s ða1 þ 1Þf ðzÞÞ0 ¼ þc ; rðzÞ þ a1 þ p pc Hp;q;s ða1 þ 1ÞgðzÞ
where
rðzÞ ¼
zðHp;q;s ða1 ÞgðzÞÞ0 : Hp;q;s ða1 ÞgðzÞ
Since gðzÞ 2 Rp ða1 þ 1; A; BÞ, from Theorem 1, we have
rðzÞ
1 þ Az ; 1 þ Bz
ð2:11Þ
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M.K. Aouf / Applied Mathematics and Computation 206 (2008) 772–780
using (1.13), we have p
rðzÞ þ a1 þ p ¼ qei2u ; where
a1 ð1 þ BÞ pðA BÞ 1þB
a1 ð1 BÞ þ pðA BÞ 1B
;
tðA; BÞ < u < tðA; BÞ;
where tðA; BÞ is given by (2.9).
Let h be a function which maps onto the angular domain w : j arg wj < p2 d with hð0Þ ¼ 1. Applying Lemma 2 for this h with kðzÞ ¼ rðzÞþ1 a1 þp, we see that Re qðzÞ > 0 in U and hence qðzÞ–0 in U. If there exists a point z0 2 U such that the conditions (2.1) and (2.2) are satisfied, then by Lemma 3, we have (2.3) under the restrictions (2.4)–(2.6). 1 At first, suppose that qðz0 Þa ¼ iaða > 0Þ. Then we obtain
arg
1 z0 ðHp;q;s ða1 þ 1Þf ðz0 ÞÞ0 z0 q0 ðz0 Þ p p þc ¼ a þ arg 1 þ iakðqei2u Þ1 ¼ arg qðz0 Þ þ pc rðz0 Þ þ a1 þ p Hp;q;s ða1 þ 1Þgðz0 Þ 2 ak sin p2 ð1 uÞ p ¼ a þ tan1 2 q þ ak cos p2 ð1 uÞ ! a sin p2 ð1 tðA; BÞÞ p p 1 P a þ tan ¼ d; a1 ð1BÞþpðABÞ 2 2 þ a cos p ð1 tðA; BÞÞ 1B
2
where d and tðA; BÞ are given by (2.8) and (2.9), respectively. This is a contradiction to the assumption of our theorem. 1 Next, suppose that pðz0 Þa ¼ iaða > 0Þ. Applying the same method as the above, we have
1 z0 ðHp;q;s ða1 þ 1Þf ðz0 ÞÞ0 p 6 a tan1 arg þc pc Hp;q;s ða1 þ 1Þgðz0 Þ 2
!
a sin p2 ð1 tðA; BÞÞ p ¼ d; a1 ð1BÞþpðABÞ p ð1 tðA; BÞÞ 2 þ a cos 1B 2
where d and tðA; BÞ are given by (2.8) and (2.9), respectively, which contradicts the assumption. Therefore we complete the proof of Theorem 2. h Taking A ¼ 1; B ¼ 0 and d ¼ 1 in Theorem 2, we have the following corollary. Corollary 1. Let f ðzÞ 2 Rp . If
zðHp;q;s ða1 þ 1Þf ðzÞÞ0 Re > c ð0 6 c < pÞ Hp;q;s ða1 þ 1ÞgðzÞ for some g 2 Rp satisfying the condition
zðHp;q;s ða1 þ 1ÞgðzÞÞ0 < p; þ p H ða þ 1ÞgðzÞ p;q;s 1 then
zðHp;q;s ða1 Þf ðzÞÞ0 Re > c ð0 6 c < pÞ: Hp;q;s ða1 ÞgðzÞ Remark 1. Taking q ¼ s þ 1; a1 ¼ b1 ¼ p; aj ¼ 1 ðj ¼ 2; 3; . . . ; s þ 1Þ and bj ¼ 1 ðj ¼ 2; . . . ; sÞ in Corollary 1 we obtain the result obtained by Lashin [10, Corollary 2.4]. Taking A ¼ 1; B ¼ 0 and gðzÞ ¼ z1p in Theorem 2, we have the following corollary. Corollary 2. Let f ðzÞ 2 Rp and a1 P p. If
j arg½zpþ1 ðHp;q;s ða1 þ 1Þf ðzÞÞ0 cj <
p 2
d ð0 6 c < p; 0 < c 6 1Þ;
then
j arg½zpþ1 ðHp;q;s ða1 Þf ðzÞÞ0 cj <
p 2
d:
Taking q ¼ s þ 1; a1 ¼ b1 ¼ p; aj ¼ 1 ðj ¼ 2; 3; . . . ; s þ 1Þ; bj ¼ 1 ðj ¼ 2; 3; . . . ; sÞ and d ¼ 1 in Corollary 2, we have the following corollary. Corollary 3. Let f ðzÞ 2 Rp . If 00
Refzpþ1 ½zf ðzÞ þ ð2p þ 1Þf 0 ðzÞg > c ð0 6 c < pÞ;
M.K. Aouf / Applied Mathematics and Computation 206 (2008) 772–780
777
then
Refzpþ1 f 0 ðzÞg > c: Remark 2. Taking q ¼ 2; s ¼ 1; a1 ¼ m; a2 ¼ 1 and b1 ¼ m þ 1; m > 1, in Theorem 2, we obtain the result obtained by Lashin [10, Corollary 2.3]. Remark 3 (i) Putting q ¼ 2; s ¼ 1; a1 ¼ n þ pðn > pÞ; a2 ¼ 1 and b1 ¼ 1 in Theorem 2, we obtain the result obtained by Cho and Owa [5, Theorem 2.1]. (ii) Putting p ¼ 1; q ¼ 2; s ¼ 1; a1 ¼ n þ 1ðn > 1Þ; a2 ¼ 1 and b1 ¼ 1 in Theorem 2, we obtain the result obtained by Cho [4, Theorem 2.1]. (iii) Putting q ¼ 2; s ¼ 1; a1 ¼ a > 0; a2 ¼ 1 and b1 ¼ c > 0 in Theorem 2, we obtain the result obtained by Lashin [10, Theorem 2.2]. By the same techniques as in the proof of Theorem 2, we obtain Theorem 3. Let f ðzÞ 2 Rp . Choose a1 2 R such that
a1 P
pðA BÞ ; 1þB
where 1 < B < A 6 1 and p 2 N. If
0 arg zðHp;q;s ða1 þ 1Þf ðzÞÞ þ c < p d ðc > p; 0 < d 6 1Þ 2 Hp;q;s ða1 þ 1ÞgðzÞ for some g 2 Rp ½a1 þ 1; A; B, then
0 arg zðHp;q;s ða1 Þf ðzÞÞ þ c < p a; 2 Hp;q;s ða1 ÞgðzÞ
where að0 < a 6 1Þ is the solution of the equation given by (2.8). Theorem 4. Let h be convex univalent in U with hð0Þ ¼ 1 and Re h be bounded in U. Let F m;p ðf ÞðzÞ be the integral operator defined by (1.10). If f 2 Rp satisfies the condition
zðHp;q;s ða1 Þf ðzÞÞ0 hðzÞ ðz 2 UÞ; pHp;q;s ða1 Þf ðzÞ
zðHp;q;s ða1 ÞF m;p ðf ÞðzÞÞ0 hðzÞ ðz 2 UÞ pHp;q;s ða1 ÞF m;p ðf ÞðzÞ
then
for maxz2U Re hðzÞ < mþp (provided Hp;q;s F m;p ðf ÞðzÞ–0 in U). p Proof. Let
pðzÞ ¼
zðHp;q;s ða1 ÞF m;p ðf ÞðzÞÞ0 pHp;q;s ða1 ÞF m;p ðf ÞðzÞ
ðz 2 UÞ:
Then, by using (1.11), we have
qðzÞ ðm þ pÞ ¼ c
Hp;q;s ða1 ÞF m;p ðf ÞðzÞ : Hp;q;s ða1 ÞF m;p ðf ÞðzÞ
Taking logarithmic derivatives in both sides of (2.12) and multiplying by z, we get
zq0 ðzÞ zðHp;q;s ða1 Þf ðzÞÞ0 þ qðzÞ ¼ hðzÞ ðz 2 UÞ: pqðzÞ þ ðm þ pÞ pHp;q;s ða1 Þf ðzÞ Therefore, by using Lemma 1, we have
zðHp;q;s ða1 ÞF m;p ðf ÞðzÞÞ0 hðzÞ ðz 2 UÞ pHp;q;s ða1 ÞF m;p ðf ÞðzÞ
for maxz2U Re hðzÞ < mþp (provided Hp;q;s ða1 ÞF m;p ðf ÞðzÞ–0 in U). h p
ð2:12Þ
778
M.K. Aouf / Applied Mathematics and Computation 206 (2008) 772–780
1þz Remark 4. Taking q ¼ 2; s ¼ 1; a1 ¼ n þ pðn > pÞ; a2 ¼ b1 ¼ 1 and hðzÞ ¼ 1z in Theorem 4, we obtain the result obtained by Aouf [1, Theorem 3].
Theorem 5. Let f ðzÞ 2 Rp and choose a positive number
mP
m such that
1þA p; 1þB
where 1 < B < A 6 1 and p 2 N. If
0 arg zðHp;q;s ða1 Þf ðzÞÞ c < p d ð0 6 c < p; 0 < d 6 1Þ 2 Hp;q;s ða1 ÞgðzÞ for some g 2 Rp ½a1 ; A; B, then
0 arg zðHp;q;s ða1 ÞF m;p ðf ÞðzÞÞ c < p a; 2 Hp;q;s ða1 ÞGm;p ðgÞðzÞ where F m;p ðf ÞðzÞ is the integral operator given by (1.10),
Gm;p ðgÞðzÞ ¼
Z
m zmþp
z
tmþp1 gðtÞdt
ðm > 0Þ
ð2:13Þ
0
and að0 < a 6 1Þ is the solution of the equation
d¼aþ
2
p
(
)
a sin p2 ð1 tðA; B; mÞÞ ; ðmþpÞð1BÞþpðABÞ þ a cos p2 ð1 tðA; B; mÞÞ 1B
1
tan
ð2:14Þ
where
tðA; B; mÞ ¼
2
p
" 1
sin
pðA BÞ ðm þ pÞð1 B2 Þ pð1 ABÞ
# ð2:15Þ
:
Proof. Let
qðzÞ ¼
1 zðHp;q;s ða1 ÞF m;p ðf ÞðzÞÞ0 þc ðz 2 UÞ: p c Hp;q;s ða1 ÞGm;p ðgÞðzÞ
Since g 2 Rp ½a1 ; A; B, from Theorem 4, Gm;p ðgÞðzÞ 2 Rp ½a1 ; A; B. Using (1.11), we have
ðp cÞqðzÞHp;q;s ða1 ÞGm;p ðgÞðzÞ ðm þ pÞHp;q;s ða1 ÞF m;p ðf ÞðzÞ ¼ mHp;q;s ða1 Þf ðzÞ cHp;q;s ða1 ÞGm;p ðgÞðzÞ: Then, by a simple calculation, we have
ðp cÞfzq0 ðzÞ þ qðzÞ½rðzÞ þ m þ pg þ c½rðzÞ þ m þ p ¼
zðHp;q;s ða1 Þf ðzÞÞ0 ; Hp;q;s ða1 ÞGm;p ðgÞðzÞ
where
rðzÞ ¼
zðHp;q;s ða1 ÞGm;p ðgÞðzÞÞ0 : Hp;q;s ða1 ÞGm;p ðgÞðzÞ
Hence, we have
qðzÞ þ
zq0 ðzÞ 1 zðHp;q;s ða1 Þf ðzÞÞ0 ¼ þc : rðzÞ þ m þ p pc Hp;q;s ða1 ÞgðzÞ
The remaining part of the proof is similar to that of Theorem 2 and so we omit it.
h
Putting q ¼ s þ 1; a1 ¼ b1 ¼ p; aj ¼ 1 ðj ¼ 2; 3; . . . ; s þ 1Þ; bj ¼ 1 ðj ¼ 2; 3; . . . ; sÞ; A ¼ 1; B ¼ 0 and d ¼ 1 in Theorem 5, we obtain the following result. Corollary 4. Let
m > 0 and f ðzÞ 2 Rp . If
0 zf ðzÞ Re > c ð0 6 c < pÞ gðzÞ for some gðzÞ 2 Rp satisfying the condition
M.K. Aouf / Applied Mathematics and Computation 206 (2008) 772–780
779
0 zg ðzÞ < p; þ p gðzÞ then
Re
( 0 ) zF m;p ðf ÞðzÞ > c ð0 6 c < pÞ; Gm;p ðgÞðzÞ
where F m;p ðf ÞðzÞ and Gm;p ðgÞðzÞ are given by (1.10) and (2.13), respectively. Taking q ¼ s þ 1; a1 ¼ b1 ¼ p; aj ¼ 1 ðj ¼ 2; 3; . . . ; s þ 1Þ; bj ¼ 1 ðj ¼ 2; 3; . . . ; sÞ; B ! A and gðzÞ ¼ z1p in Theorem 5, we obtain the following corollary. Corollary 5. Let
j argðz
m > 0 and f ðzÞ 2 Rp . If
p
pþ1 0
f ðzÞ cÞj <
2
d ð0 6 c < p; 0 < d 6 1Þ
then
j argðzpþ1 F 0m;p ðf ÞðzÞ cÞj <
p 2
a;
where F m;p ðf ÞðzÞ is the integral operator given by (1.10) and að0 < a 6 1Þ is the solution of the equation
d¼aþ
2
p
tan1
a
mþp
:
By using the same method as in proving Theorem 5, we have Theorem 6. Let f ðzÞ 2 Rp and choose a positive number
m such that
1þA mP p; 1þB where 1 < B < A 6 1 and p 2 N. If
0 arg zðHp;q;s ða1 Þf ðzÞÞ þ c < p d ðc > p; 0 < d 6 1Þ 2 Hp;q;s ða1 ÞgðzÞ for some g 2 Rp ½a1 ; A; B, then
0 arg zðHp;q;s ða1 ÞF m;p ðf ÞðzÞÞ þ c < p a; 2 Hp;q;s ða1 ÞGm;p ðgÞðzÞ
where F m;p ðf ÞðzÞ and Gm;p ðgÞðzÞ are given by (1.10) and (2.13), respectively, and að0 < a 6 1Þ is the solution of the equation given by (2.14). Finally, we derive Theorem 7. Let f ðzÞ 2 Rp . Choose a1 2 R such that
a1 P
pðA BÞ ; 1þB
where 1 < B < A 6 1 and p 2 N. If
0 arg zðHp;q;s ða1 Þf ðzÞÞ c < p d ð0 6 c < p; 0 < d 6 1Þ 2 Hp;q;s ða1 ÞgðzÞ for some g 2 Rp ½a1 ; A; B, then
0 arg zðHp;q;s ða1 þ 1ÞF m;p ðf ÞðzÞÞ c < p d; 2 Hp;q;s ða1 þ 1ÞGm;p ðgÞðzÞ
where F m;p ðf ÞðzÞ and Gm;p ðgÞðzÞ are given by (1.10) and (2.13), respectively, with Proof. From (1.9) and (1.11) with
m ¼ a1 , we have Hp;q;s ða1 Þf ðzÞ ¼ Hp;q;s ða1 þ 1ÞF m;p ðf ÞðzÞ. Therefore
zðHp;q;s ða1 Þf ðzÞÞ0 zðHp;q;s ða1 þ 1ÞF m;p ðf ÞðzÞÞ0 ¼ Hp;q;s ða1 ÞgðzÞ Hp;q;s ða1 þ 1ÞGm;p ðgÞðzÞ and the result follows. h
m ¼ a1 .
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M.K. Aouf / Applied Mathematics and Computation 206 (2008) 772–780
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