- Email: [email protected]
Subordinations for analytic functions defined by the Dziok–Srivastava linear operator
Applied Mathematics and Computation 187 (2007) 13–19 www.elsevier.com/locate/amc
Subordinations for analytic functions defined by the Dziok–Srivastava linear operator R. Aghalary
a,*
, S.B. Joshi b, R.N. Mohapatra c, V. Ravichandran
d
a
b
Department of Mathematics, University of Urmia, Urmia, Iran Department of Mathematics, Walchand College of Engineering, Sangli 416415, Maharashtra, India c Department of Mathematics, University of Central Florida, Orlando, FL 32816, USA d School of Mathematical Sciences, Universiti Sains Malaysia, 11800 USM Penang, Malaysia
Dedicated to Professor H.M. Srivastava on the occasion of his 65th birthday
Abstract In the present investigation, we obtain certain sufficient conditions for a normalized analytic function f(z) defined by the Dziok–Srivastava linear operator H lm ½a1 to satisfy the certain subordination. Our results extend corresponding previously known results on starlikeness, convexity, and close to convexity. Ó 2006 Elsevier Inc. All rights reserved. Keywords: Univalent functions; Starlike functions; Convex functions; Differential subordination; Convolution; Dziok–Srivastava linear operator
1. Introduction Let H denote the class of analytic functions defined on the open unit disc D ¼ fz 2 C :j z j< 1g. Let A denote the subclass of H consisting of functions f(z) normalized by f(0) = f 0 (0) 1 = 0. For the functions f and g in H, we say that f is subordinate to g in D, and write f g, if there exists a Schwarz P function x in 1 k H with jx(z)j < 1 and x(0) = 0 such that f(z) = g(x(z)) in D. For two functions f ðzÞ ¼ z þ k¼2 ak z and P1 gðzÞ ¼ z þ k¼2 bk zk , the Hadamard product (or convolution) of f and g is defined by 1 X ak bk zk ¼: ðg f ÞðzÞ: ðf gÞðzÞ :¼ z þ k¼2
*
Corresponding author. E-mail addresses: [email protected] (R. Aghalary), [email protected] (S.B. Joshi), [email protected] (R.N. Mohapatra), [email protected] (V. Ravichandran). URL: http://cs.usm.my/~vravi (V. Ravichandran). 0096-3003/$ - see front matter Ó 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2006.08.097
14
R. Aghalary et al. / Applied Mathematics and Computation 187 (2007) 13–19
For aj 2 Cðj ¼ 1; 2; . . . ; lÞ and bj 2 C n f0; 1; 2; . . .gðj ¼ 1; 2; . . . ; mÞ; the generalized hypergeometric function lFm(a1, . . . , al;b1, . . . , bm;z) is defined by the infinite series 1 X ða1 Þn ðal Þn zn ðl 6 m þ 1; l; m 2 N0 :¼ f0; 1; 2; . . .gÞ; l F m ða1 ; . . . ; al ; b1 ; . . . ; bm ; zÞ :¼ ðb1 Þn ðbm Þn n! n¼0 where (a)n is the Pochhammer symbol defined by 1; ðn ¼ 0Þ; Cða þ nÞ ðaÞn :¼ ¼ CðaÞ aða þ 1Þða þ 2Þ ða þ n 1Þ; ðn 2 N :¼ f1; 2; 3; . . . gÞ: Corresponding to the function hða1 ; . . . ; al ; b1 ; . . . ; bm ; zÞ :¼ zl F m ða1 ; . . . ; al ; b1 ; . . . ; bm ; zÞ; the Dziok–Srivastava operator [3] (see also [4,13]) H(l,m)(a1, . . . , al; b1, . . . , bm) is defined by the Hadamard product
H ðl;mÞ ða1 ; . . . ; al ; b1 ; . . . ; bm Þf ðzÞ :¼ hða1 ; . . . ; al ; b1 ; . . . ; bm ; zÞ f ðzÞ 1 X ða1 Þn1 . . . ðal Þn1 an zn ¼zþ : ðb1 Þn1 . . . ðbm Þn1 ðn 1Þ! n¼2
ð1:1Þ
For brevity, we write H lm ½a1 f ðzÞ :¼ H ðl;mÞ ða1 ; . . . ; al ; b1 ; . . . ; bm Þf ðzÞ: Special cases of the Dziok–Srivastava linear operator includes the Hohlov linear operator [5], the Carlson– Shaffer linear operator L(a, c) [2], the Ruscheweyh derivative operator Dn [12], the generalized Bernardi–Libera–Livingston linear integral operator (cf. [1,6,7]), and the Srivastava–Owa fractional derivative operators (cf. [9,10]). In the present paper, we obtain certain sufficient conditions for a function f 2 A to satisfy either of the following subordinations: H lm ½a1 þ 1f ðzÞ kð1 zÞ ; kz H lm ½a1 f ðzÞ
H lm ½a1 f ðzÞ 1 þ Az ; z 1z
H lm ½a1 f ðzÞ kð1 zÞ : z kz
Our results extend corresponding previously known results on starlikeness, convexity, and close to convexity. To prove our main results, we need the following: Lemma 1.1 (cf. Miller and Mocanu [8, Theorem 3.4h, p.132]). Let q(z) be univalent in the unit disk D and let # and u be analytic in a domain D q(D), with u(w) 5 0 when w 2 q(D). Set QðzÞ :¼ zq0 ðzÞuðqðzÞÞ;
hðzÞ :¼ #ðqðzÞÞ þ QðzÞ:
Suppose that (1) Q(z) is starlike univalent in D, and (2) R zh0ðzÞ > 0 for z 2 D. QðzÞ If p(z) is analytic in D with p(0) = q(0), p(D) D and #ðpðzÞÞ þ zp0 ðzÞuðpðzÞÞ #ðqðzÞÞ þ zq0 ðzÞuðqðzÞÞ; then p(z) q(z) and q(z) is the best dominant. 2. Main results We begin with the following:
ð1:2Þ
R. Aghalary et al. / Applied Mathematics and Computation 187 (2007) 13–19
Theorem 2.1. Let a1 P 0, a 2 R satisfy jaj 6 1 and k > 1. If f 2 A satisfies H lm ½a1 f ðzÞ=z 6¼ 0 in D and l a H m ½a1 þ 1f ðzÞ H lm ½a1 þ 2f ðzÞ 1 hðzÞ; ða þ 1Þ 1 H lm ½a1 f ðzÞ H lm ½a1 þ 1f ðzÞ
15
ð2:1Þ
where
! 1þa kð1 zÞ ðk 1Þz hðzÞ ¼ a1 ; 2 kz kð1 zÞ
then H lm ½a1 þ 1f ðzÞ kð1 zÞ : kz H lm ½a1 f ðzÞ Proof. The condition (2.1) and H lm ½a1 f ðzÞ=z 6¼ 0 in D implies that H lm ½a1 þ 1f ðzÞ=z 6¼ 0 in D. Define the function p(z) by pðzÞ :¼
H lm ½a1 þ 1f ðzÞ : H lm ½a1 f ðzÞ
Clearly p(z) is analytic in D. A computation shows that 0
0
zp0 ðzÞ z½H lm ½a1 þ 1f ðzÞ z½H lm ½a1 f ðzÞ ¼ : pðzÞ H lm ½a1 þ 1f ðzÞ H lm ½a1 f ðzÞ
ð2:2Þ
By using the identity z½H lm ½a1 f ðzÞ0 ¼ a1 H lm ½a1 þ 1f ðzÞ ða1 1ÞH lm ½a1 f ðzÞ;
ð2:3Þ
we get, from (2.2), ða1 þ 1Þ
H lm ½a1 þ 2f ðzÞ zp0 ðzÞ : ¼ 1 þ a1 pðzÞ þ l pðzÞ H m ½a1 þ 1f ðzÞ
ð2:4Þ
Using (2.4) in (2.1), we get a1 ðpðzÞÞ
aþ1
a1
þ zp0 ðzÞðpðzÞÞ
hðzÞ:
ð2:5Þ
Let q(z) be the function defined by qðzÞ :¼
kð1 zÞ : kz
It is clear that q is convex univalent in D. Since hðzÞ ¼ a1 ðqðzÞÞaþ1 þ zq0 ðzÞðqðzÞÞa1 ; we see that (2.5) can be written as (1.2) when # and u are given by #ðwÞ ¼ a1 waþ1
and
uðwÞ ¼ wa1 :
Clearly u and # are analytic in C n f0g. Now a1
QðzÞ ¼ zq0 ðzÞuðqðzÞÞ ¼ zq0 ðzÞðqðzÞÞ hðzÞ ¼ #ðqðzÞÞ þ QðzÞ ¼
kð1 zÞ kz
¼
ð1 kÞzka ð1 zÞa1 1þa
ðk zÞ
1þa a1
ðk 1Þz kð1 zÞ
2
! :
;
16
R. Aghalary et al. / Applied Mathematics and Computation 187 (2007) 13–19
By our assumptions on the parameters a and k, we see that 0 zQ ðzÞ zð1 aÞ z þ ð1 þ aÞ R ¼R 1þ QðzÞ 1z kz 1 ð1 þ aÞk > 1 þ ð1 aÞ þ 2 1þk ð1 þ aÞðk 1Þ ¼ > 0; 2ð1 þ kÞ and therefore Q(z) is starlike. Also we have R
zh0 ðzÞ kð1 zÞ zQ0 ðzÞ ¼ a1 ð1 þ aÞR þR P 0: QðzÞ kz QðzÞ
By an application of Lemma 1.1, we have p(z) q(z) or H lm ½a1 þ 1f ðzÞ kð1 zÞ : kz H lm ½a1 f ðzÞ
Since 1 H 10 ½3f ðzÞ ¼ z2 f 00 ðzÞ þ zf 0 ðzÞ; 2 by taking a = 0, l = 1, m = 0 and a1 = 1 in Theorem 2.1, we get the following corollary: H 10 ½1f ðzÞ ¼ f ðzÞ; H 10 ½2f ðzÞ ¼ zf 0 ðzÞ;
Corollary 2.1. Let f 2 A be so that f(z)/z 5 0 in D. If k > 1 and 1þ
zf 00 ðzÞ kð1 zÞ ðk 1Þz ; f 0 ðzÞ kz ð1 zÞðk zÞ
then zf 0 ðzÞ kð1 zÞ : f ðzÞ kz Remark 2.1. The function hðzÞ ¼
kð1 zÞ ðk 1Þz z 1 ¼ þ ðk zÞ ð1 zÞðk zÞ kz 1z
ðkþ1Þ ð5k1Þ takes real value for real value of z,h(0) = 1 and h(D) is the region RhðzÞ < 2ðk1Þ for 1 < k 6 2 and RhðzÞ < 2ðkþ1Þ for 2 < k. Hence this result generalizes the result obtained by Owa et al. [11].
We note that the image of the function hðzÞ ¼ 1
ðk 1Þz kð1 zÞ
is
hðDÞ ¼ C
2
5k 1 ;1 : 4k
Hence by taking a = 1, a1 = 1, l = 1, and m = 0 in Theorem 2.1, we get the following corollary: Corollary 2.2. Let k > 1, f 2 A and f(z)/z 5 0 in D. If f satisfies 0 1 00 ðzÞ 1 þ zff 0 ðzÞ 5k 1 ; R@ zf 0 ðzÞ A < 4k f ðzÞ
R. Aghalary et al. / Applied Mathematics and Computation 187 (2007) 13–19
17
then zf 0 ðzÞ kð1 zÞ : f ðzÞ kz Theorem 2.2. Let a1 > 0, 1 6 a < 0, 1 < A < 1. If f 2 A satisfy the condition H lm ½a1 f ðzÞ=z 6¼ 0 in D and l a H m ½a1 f ðzÞ H l ½a1 þ 1f ðzÞ a1 m hðzÞ; z z
ð2:6Þ
where ! a 1 þ Az 1 þ Az ð1 þ AÞz hðzÞ ¼ þ a1 ; 2 1z 1z ð1 zÞ then H lm ½a1 f ðzÞ 1 þ Az : z 1z Proof. Define the function p(z) by pðzÞ :¼
H lm ½a1 f ðzÞ : z
ð2:7Þ
It is clear that p(0) = 1 and p is analytic in D. By using the identity (2.3), we get, from (2.7), a1 H lm ½a1 þ 1f ðzÞ ¼ zp0 ðzÞ þ a1 pðzÞ:
ð2:8Þ
Using (2.8) in (2.6), we see that the subordination becomes a1 pðzÞ
1þa
a
þ pðzÞ zp0 ðzÞ hðzÞ:
Define the function q(z) by qðzÞ :¼
1 þ Az : 1z
It is clear that q(z) is univalent in D and q(D) is the region RqðzÞ > ð1 AÞ=2. By defining the functions # and u by #ðwÞ ¼ a1 w1þa
and
uðwÞ ¼ wa ;
we observe that (2.6) can be written as (1.2). Note that u and # are analytic in C n f0g. Also we see that QðzÞ :¼ zq0 ðzÞuðqðzÞÞ ¼ and
ð1 þ AÞzð1 þ AzÞ ð1 zÞ
2þa
a
;
! a 1 þ Az 1 þ Az ð1 þ AÞz þ hðzÞ :¼ #ðqðzÞÞ þ QðzÞ ¼ a1 : 2 1z 1z ð1 zÞ
By our assumptions, we have zQ0 ðzÞ Az z ajAj 2 þ a að1 jAjÞ ¼R 1þa þ ð2 þ aÞ ¼ > 0; R >1 QðzÞ 1 þ Az 1z 1 þ jAj 2 2ð1 þ jAjÞ
18
and
R. Aghalary et al. / Applied Mathematics and Computation 187 (2007) 13–19
0 zh0 ðzÞ # ðqðzÞÞ zQ0 ðzÞ zQ0 ðzÞ ¼R þ P 0: R ¼ a1 ð1 þ aÞ þ R QðzÞ uðqðzÞÞ QðzÞ QðzÞ
The results now follows by an application of Lemma 1.1. ð1þAÞz ð1zÞð1þAzÞ
The function hðzÞ ¼ a1 þ with respect to the real axis and 1 1 ; RhðzÞ > a1 þ 2 1 jAj
h
takes real values for real values of z with h(0) = a1 and h(D) is symmetric
z 2 D:
Consequently, by letting a = 1 in Theorem 2.2, we obtain the following corollary, which is extension to [11, Theorem 1, p. 64] which does not extend as for the sharpness. Corollary 2.3. Let 1 < A < 1, a1 > 0 and f 2 A be so that H lm ½a1 f ðzÞ=z 6¼ 0 in D and R
H lm ½a1 þ 1f ðzÞ 1 1 ; >1þ 2a1 a1 ð1 jAjÞ H lm ½a1 f ðzÞ
then H lm ½a1 f ðzÞ 1 þ Az : z 1z Theorem 2.3. Let a P 1, k > 1, f 2 A and H lm ½a1 f ðzÞ 6¼ 0 for 0 < jzj < 1. If f satisfies
H lm ½a1 f ðzÞ z
a a H lm ½a1 þ 1f ðzÞ k1þa ð1 zÞ ðk 1Þz a1 a1 ð1 zÞ ; z kz ðk zÞ1þa
then H lm ½a1 f ðzÞ kð1 zÞ : z kz Proof. The proof of Theorem 2.3, also based upon Lemma 1.1 is similar to that of Theorem 2.1. Indeed, in this case, the results follows from Lemma 1.1 when we define the functions u and # by #(w) = a1w(1+a) and u(w) = w(2+a). h ðk1Þz 3k1 Finally we note that R 1 ðkzÞð1zÞ for z 2 D and so from above Theorem by choosing l = 1, < 2ðk1Þ m = 0, and a1 = 1 we can get the following corollary: Corollary 2.4. Let k > 1, f 2 A and f 0 (z) 5 0 in D. If f satisfies
zf 00 ðzÞ R 1þ 0 f ðzÞ
<
3k 1 ; 2ðk 1Þ
then f 0 ðzÞ
kð1 zÞ : kz
Acknowledgement The research of V. Ravichandran is supported by a post-doctoral research fellowship from Universiti Sains Malaysia.
R. Aghalary et al. / Applied Mathematics and Computation 187 (2007) 13–19
19
References [1] S.D. Bernardi, Convex and starlike univalent functions, Trans. Am. Math. Soc. 135 (1969) 429–446. [2] B.C. Carlson, D.B. Shaffer, Starlike and prestarlike hypergeometric functions, SIAM J. Math. Anal. 15 (4) (1984) 737–745. [3] J. Dziok, H.M. Srivastava, Classes of analytic functions associated with the generalized hypergeometric function, Appl. Math. Comput. 103 (1999) 1–13. [4] J. Dziok, H.M. Srivastava, Certain subclasses of analytic functions associated with the generalized hypergeometric function, Integral Transform. Spec. Funct. 14 (2003) 7–18. [5] Ju.E. Hohlov, Operators and operations on the class of univalent functions, Izv. Vyssh. Uchebn. Zaved. Mat. 10 (197) (1978) 83–89. [6] R.J. Libera, Some classes of regular univalent functions, Proc. Am. Math. Soc. 16 (1965) 755–758. [7] A.E. Livingston, On the radius of univalence of certain analytic functions, Proc. Am. Math. Soc. 17 (1966) 352–357. [8] S.S. Miller, P.T. Mocanu, Differential subordinations: theory and applications, Series in Pure and Applied Mathematics, 225, Marcel Dekker, New York, 2000. [9] S. Owa, On the distortion theorems, I, Kyungpook Math. J. 18 (1978) 53–59. [10] S. Owa, H.M. Srivastava, Univalent and starlike generalized hypergeometric functions, Canad. J. Math. 39 (1987) 1057–1077. [11] S. Owa, M. Nunokawa, H. Saitoh, H.M. Srivastava, Close-to-convexity, starlikeness, and convexity of certain analytic functions, Appl. Math. Lett. 15 (1) (2002) 63–69. [12] S. Ruscheweyh, New criteria for univalent functions, Proc. Am. Math. Soc. 49 (1975) 109–115. [13] H.M. Srivastava, Some families of fractional derivative and other linear operators associated with analytic, univalent and multivalent functions, in: K.S. Lakshmi (Ed.), Proceedings of the International Conference on Analysis and Its Applications, Allied Publishers Limited, New Delhi and Chennai, 2001, pp. 209–243.
Subordinations for analytic functions defined by the Dziok–Srivastava linear operator R. Aghalary
a,*
, S.B. Joshi b, R.N. Mohapatra c, V. Ravichandran
d
a
b
Department of Mathematics, University of Urmia, Urmia, Iran Department of Mathematics, Walchand College of Engineering, Sangli 416415, Maharashtra, India c Department of Mathematics, University of Central Florida, Orlando, FL 32816, USA d School of Mathematical Sciences, Universiti Sains Malaysia, 11800 USM Penang, Malaysia
Dedicated to Professor H.M. Srivastava on the occasion of his 65th birthday
Abstract In the present investigation, we obtain certain sufficient conditions for a normalized analytic function f(z) defined by the Dziok–Srivastava linear operator H lm ½a1 to satisfy the certain subordination. Our results extend corresponding previously known results on starlikeness, convexity, and close to convexity. Ó 2006 Elsevier Inc. All rights reserved. Keywords: Univalent functions; Starlike functions; Convex functions; Differential subordination; Convolution; Dziok–Srivastava linear operator
1. Introduction Let H denote the class of analytic functions defined on the open unit disc D ¼ fz 2 C :j z j< 1g. Let A denote the subclass of H consisting of functions f(z) normalized by f(0) = f 0 (0) 1 = 0. For the functions f and g in H, we say that f is subordinate to g in D, and write f g, if there exists a Schwarz P function x in 1 k H with jx(z)j < 1 and x(0) = 0 such that f(z) = g(x(z)) in D. For two functions f ðzÞ ¼ z þ k¼2 ak z and P1 gðzÞ ¼ z þ k¼2 bk zk , the Hadamard product (or convolution) of f and g is defined by 1 X ak bk zk ¼: ðg f ÞðzÞ: ðf gÞðzÞ :¼ z þ k¼2
*
Corresponding author. E-mail addresses: [email protected] (R. Aghalary), [email protected] (S.B. Joshi), [email protected] (R.N. Mohapatra), [email protected] (V. Ravichandran). URL: http://cs.usm.my/~vravi (V. Ravichandran). 0096-3003/$ - see front matter Ó 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2006.08.097
14
R. Aghalary et al. / Applied Mathematics and Computation 187 (2007) 13–19
For aj 2 Cðj ¼ 1; 2; . . . ; lÞ and bj 2 C n f0; 1; 2; . . .gðj ¼ 1; 2; . . . ; mÞ; the generalized hypergeometric function lFm(a1, . . . , al;b1, . . . , bm;z) is defined by the infinite series 1 X ða1 Þn ðal Þn zn ðl 6 m þ 1; l; m 2 N0 :¼ f0; 1; 2; . . .gÞ; l F m ða1 ; . . . ; al ; b1 ; . . . ; bm ; zÞ :¼ ðb1 Þn ðbm Þn n! n¼0 where (a)n is the Pochhammer symbol defined by 1; ðn ¼ 0Þ; Cða þ nÞ ðaÞn :¼ ¼ CðaÞ aða þ 1Þða þ 2Þ ða þ n 1Þ; ðn 2 N :¼ f1; 2; 3; . . . gÞ: Corresponding to the function hða1 ; . . . ; al ; b1 ; . . . ; bm ; zÞ :¼ zl F m ða1 ; . . . ; al ; b1 ; . . . ; bm ; zÞ; the Dziok–Srivastava operator [3] (see also [4,13]) H(l,m)(a1, . . . , al; b1, . . . , bm) is defined by the Hadamard product
H ðl;mÞ ða1 ; . . . ; al ; b1 ; . . . ; bm Þf ðzÞ :¼ hða1 ; . . . ; al ; b1 ; . . . ; bm ; zÞ f ðzÞ 1 X ða1 Þn1 . . . ðal Þn1 an zn ¼zþ : ðb1 Þn1 . . . ðbm Þn1 ðn 1Þ! n¼2
ð1:1Þ
For brevity, we write H lm ½a1 f ðzÞ :¼ H ðl;mÞ ða1 ; . . . ; al ; b1 ; . . . ; bm Þf ðzÞ: Special cases of the Dziok–Srivastava linear operator includes the Hohlov linear operator [5], the Carlson– Shaffer linear operator L(a, c) [2], the Ruscheweyh derivative operator Dn [12], the generalized Bernardi–Libera–Livingston linear integral operator (cf. [1,6,7]), and the Srivastava–Owa fractional derivative operators (cf. [9,10]). In the present paper, we obtain certain sufficient conditions for a function f 2 A to satisfy either of the following subordinations: H lm ½a1 þ 1f ðzÞ kð1 zÞ ; kz H lm ½a1 f ðzÞ
H lm ½a1 f ðzÞ 1 þ Az ; z 1z
H lm ½a1 f ðzÞ kð1 zÞ : z kz
Our results extend corresponding previously known results on starlikeness, convexity, and close to convexity. To prove our main results, we need the following: Lemma 1.1 (cf. Miller and Mocanu [8, Theorem 3.4h, p.132]). Let q(z) be univalent in the unit disk D and let # and u be analytic in a domain D q(D), with u(w) 5 0 when w 2 q(D). Set QðzÞ :¼ zq0 ðzÞuðqðzÞÞ;
hðzÞ :¼ #ðqðzÞÞ þ QðzÞ:
Suppose that (1) Q(z) is starlike univalent in D, and (2) R zh0ðzÞ > 0 for z 2 D. QðzÞ If p(z) is analytic in D with p(0) = q(0), p(D) D and #ðpðzÞÞ þ zp0 ðzÞuðpðzÞÞ #ðqðzÞÞ þ zq0 ðzÞuðqðzÞÞ; then p(z) q(z) and q(z) is the best dominant. 2. Main results We begin with the following:
ð1:2Þ
R. Aghalary et al. / Applied Mathematics and Computation 187 (2007) 13–19
Theorem 2.1. Let a1 P 0, a 2 R satisfy jaj 6 1 and k > 1. If f 2 A satisfies H lm ½a1 f ðzÞ=z 6¼ 0 in D and l a H m ½a1 þ 1f ðzÞ H lm ½a1 þ 2f ðzÞ 1 hðzÞ; ða þ 1Þ 1 H lm ½a1 f ðzÞ H lm ½a1 þ 1f ðzÞ
15
ð2:1Þ
where
! 1þa kð1 zÞ ðk 1Þz hðzÞ ¼ a1 ; 2 kz kð1 zÞ
then H lm ½a1 þ 1f ðzÞ kð1 zÞ : kz H lm ½a1 f ðzÞ Proof. The condition (2.1) and H lm ½a1 f ðzÞ=z 6¼ 0 in D implies that H lm ½a1 þ 1f ðzÞ=z 6¼ 0 in D. Define the function p(z) by pðzÞ :¼
H lm ½a1 þ 1f ðzÞ : H lm ½a1 f ðzÞ
Clearly p(z) is analytic in D. A computation shows that 0
0
zp0 ðzÞ z½H lm ½a1 þ 1f ðzÞ z½H lm ½a1 f ðzÞ ¼ : pðzÞ H lm ½a1 þ 1f ðzÞ H lm ½a1 f ðzÞ
ð2:2Þ
By using the identity z½H lm ½a1 f ðzÞ0 ¼ a1 H lm ½a1 þ 1f ðzÞ ða1 1ÞH lm ½a1 f ðzÞ;
ð2:3Þ
we get, from (2.2), ða1 þ 1Þ
H lm ½a1 þ 2f ðzÞ zp0 ðzÞ : ¼ 1 þ a1 pðzÞ þ l pðzÞ H m ½a1 þ 1f ðzÞ
ð2:4Þ
Using (2.4) in (2.1), we get a1 ðpðzÞÞ
aþ1
a1
þ zp0 ðzÞðpðzÞÞ
hðzÞ:
ð2:5Þ
Let q(z) be the function defined by qðzÞ :¼
kð1 zÞ : kz
It is clear that q is convex univalent in D. Since hðzÞ ¼ a1 ðqðzÞÞaþ1 þ zq0 ðzÞðqðzÞÞa1 ; we see that (2.5) can be written as (1.2) when # and u are given by #ðwÞ ¼ a1 waþ1
and
uðwÞ ¼ wa1 :
Clearly u and # are analytic in C n f0g. Now a1
QðzÞ ¼ zq0 ðzÞuðqðzÞÞ ¼ zq0 ðzÞðqðzÞÞ hðzÞ ¼ #ðqðzÞÞ þ QðzÞ ¼
kð1 zÞ kz
¼
ð1 kÞzka ð1 zÞa1 1þa
ðk zÞ
1þa a1
ðk 1Þz kð1 zÞ
2
! :
;
16
R. Aghalary et al. / Applied Mathematics and Computation 187 (2007) 13–19
By our assumptions on the parameters a and k, we see that 0 zQ ðzÞ zð1 aÞ z þ ð1 þ aÞ R ¼R 1þ QðzÞ 1z kz 1 ð1 þ aÞk > 1 þ ð1 aÞ þ 2 1þk ð1 þ aÞðk 1Þ ¼ > 0; 2ð1 þ kÞ and therefore Q(z) is starlike. Also we have R
zh0 ðzÞ kð1 zÞ zQ0 ðzÞ ¼ a1 ð1 þ aÞR þR P 0: QðzÞ kz QðzÞ
By an application of Lemma 1.1, we have p(z) q(z) or H lm ½a1 þ 1f ðzÞ kð1 zÞ : kz H lm ½a1 f ðzÞ
Since 1 H 10 ½3f ðzÞ ¼ z2 f 00 ðzÞ þ zf 0 ðzÞ; 2 by taking a = 0, l = 1, m = 0 and a1 = 1 in Theorem 2.1, we get the following corollary: H 10 ½1f ðzÞ ¼ f ðzÞ; H 10 ½2f ðzÞ ¼ zf 0 ðzÞ;
Corollary 2.1. Let f 2 A be so that f(z)/z 5 0 in D. If k > 1 and 1þ
zf 00 ðzÞ kð1 zÞ ðk 1Þz ; f 0 ðzÞ kz ð1 zÞðk zÞ
then zf 0 ðzÞ kð1 zÞ : f ðzÞ kz Remark 2.1. The function hðzÞ ¼
kð1 zÞ ðk 1Þz z 1 ¼ þ ðk zÞ ð1 zÞðk zÞ kz 1z
ðkþ1Þ ð5k1Þ takes real value for real value of z,h(0) = 1 and h(D) is the region RhðzÞ < 2ðk1Þ for 1 < k 6 2 and RhðzÞ < 2ðkþ1Þ for 2 < k. Hence this result generalizes the result obtained by Owa et al. [11].
We note that the image of the function hðzÞ ¼ 1
ðk 1Þz kð1 zÞ
is
hðDÞ ¼ C
2
5k 1 ;1 : 4k
Hence by taking a = 1, a1 = 1, l = 1, and m = 0 in Theorem 2.1, we get the following corollary: Corollary 2.2. Let k > 1, f 2 A and f(z)/z 5 0 in D. If f satisfies 0 1 00 ðzÞ 1 þ zff 0 ðzÞ 5k 1 ; R@ zf 0 ðzÞ A < 4k f ðzÞ
R. Aghalary et al. / Applied Mathematics and Computation 187 (2007) 13–19
17
then zf 0 ðzÞ kð1 zÞ : f ðzÞ kz Theorem 2.2. Let a1 > 0, 1 6 a < 0, 1 < A < 1. If f 2 A satisfy the condition H lm ½a1 f ðzÞ=z 6¼ 0 in D and l a H m ½a1 f ðzÞ H l ½a1 þ 1f ðzÞ a1 m hðzÞ; z z
ð2:6Þ
where ! a 1 þ Az 1 þ Az ð1 þ AÞz hðzÞ ¼ þ a1 ; 2 1z 1z ð1 zÞ then H lm ½a1 f ðzÞ 1 þ Az : z 1z Proof. Define the function p(z) by pðzÞ :¼
H lm ½a1 f ðzÞ : z
ð2:7Þ
It is clear that p(0) = 1 and p is analytic in D. By using the identity (2.3), we get, from (2.7), a1 H lm ½a1 þ 1f ðzÞ ¼ zp0 ðzÞ þ a1 pðzÞ:
ð2:8Þ
Using (2.8) in (2.6), we see that the subordination becomes a1 pðzÞ
1þa
a
þ pðzÞ zp0 ðzÞ hðzÞ:
Define the function q(z) by qðzÞ :¼
1 þ Az : 1z
It is clear that q(z) is univalent in D and q(D) is the region RqðzÞ > ð1 AÞ=2. By defining the functions # and u by #ðwÞ ¼ a1 w1þa
and
uðwÞ ¼ wa ;
we observe that (2.6) can be written as (1.2). Note that u and # are analytic in C n f0g. Also we see that QðzÞ :¼ zq0 ðzÞuðqðzÞÞ ¼ and
ð1 þ AÞzð1 þ AzÞ ð1 zÞ
2þa
a
;
! a 1 þ Az 1 þ Az ð1 þ AÞz þ hðzÞ :¼ #ðqðzÞÞ þ QðzÞ ¼ a1 : 2 1z 1z ð1 zÞ
By our assumptions, we have zQ0 ðzÞ Az z ajAj 2 þ a að1 jAjÞ ¼R 1þa þ ð2 þ aÞ ¼ > 0; R >1 QðzÞ 1 þ Az 1z 1 þ jAj 2 2ð1 þ jAjÞ
18
and
R. Aghalary et al. / Applied Mathematics and Computation 187 (2007) 13–19
0 zh0 ðzÞ # ðqðzÞÞ zQ0 ðzÞ zQ0 ðzÞ ¼R þ P 0: R ¼ a1 ð1 þ aÞ þ R QðzÞ uðqðzÞÞ QðzÞ QðzÞ
The results now follows by an application of Lemma 1.1. ð1þAÞz ð1zÞð1þAzÞ
The function hðzÞ ¼ a1 þ with respect to the real axis and 1 1 ; RhðzÞ > a1 þ 2 1 jAj
h
takes real values for real values of z with h(0) = a1 and h(D) is symmetric
z 2 D:
Consequently, by letting a = 1 in Theorem 2.2, we obtain the following corollary, which is extension to [11, Theorem 1, p. 64] which does not extend as for the sharpness. Corollary 2.3. Let 1 < A < 1, a1 > 0 and f 2 A be so that H lm ½a1 f ðzÞ=z 6¼ 0 in D and R
H lm ½a1 þ 1f ðzÞ 1 1 ; >1þ 2a1 a1 ð1 jAjÞ H lm ½a1 f ðzÞ
then H lm ½a1 f ðzÞ 1 þ Az : z 1z Theorem 2.3. Let a P 1, k > 1, f 2 A and H lm ½a1 f ðzÞ 6¼ 0 for 0 < jzj < 1. If f satisfies
H lm ½a1 f ðzÞ z
a a H lm ½a1 þ 1f ðzÞ k1þa ð1 zÞ ðk 1Þz a1 a1 ð1 zÞ ; z kz ðk zÞ1þa
then H lm ½a1 f ðzÞ kð1 zÞ : z kz Proof. The proof of Theorem 2.3, also based upon Lemma 1.1 is similar to that of Theorem 2.1. Indeed, in this case, the results follows from Lemma 1.1 when we define the functions u and # by #(w) = a1w(1+a) and u(w) = w(2+a). h ðk1Þz 3k1 Finally we note that R 1 ðkzÞð1zÞ for z 2 D and so from above Theorem by choosing l = 1, < 2ðk1Þ m = 0, and a1 = 1 we can get the following corollary: Corollary 2.4. Let k > 1, f 2 A and f 0 (z) 5 0 in D. If f satisfies
zf 00 ðzÞ R 1þ 0 f ðzÞ
<
3k 1 ; 2ðk 1Þ
then f 0 ðzÞ
kð1 zÞ : kz
Acknowledgement The research of V. Ravichandran is supported by a post-doctoral research fellowship from Universiti Sains Malaysia.
R. Aghalary et al. / Applied Mathematics and Computation 187 (2007) 13–19
19
References [1] S.D. Bernardi, Convex and starlike univalent functions, Trans. Am. Math. Soc. 135 (1969) 429–446. [2] B.C. Carlson, D.B. Shaffer, Starlike and prestarlike hypergeometric functions, SIAM J. Math. Anal. 15 (4) (1984) 737–745. [3] J. Dziok, H.M. Srivastava, Classes of analytic functions associated with the generalized hypergeometric function, Appl. Math. Comput. 103 (1999) 1–13. [4] J. Dziok, H.M. Srivastava, Certain subclasses of analytic functions associated with the generalized hypergeometric function, Integral Transform. Spec. Funct. 14 (2003) 7–18. [5] Ju.E. Hohlov, Operators and operations on the class of univalent functions, Izv. Vyssh. Uchebn. Zaved. Mat. 10 (197) (1978) 83–89. [6] R.J. Libera, Some classes of regular univalent functions, Proc. Am. Math. Soc. 16 (1965) 755–758. [7] A.E. Livingston, On the radius of univalence of certain analytic functions, Proc. Am. Math. Soc. 17 (1966) 352–357. [8] S.S. Miller, P.T. Mocanu, Differential subordinations: theory and applications, Series in Pure and Applied Mathematics, 225, Marcel Dekker, New York, 2000. [9] S. Owa, On the distortion theorems, I, Kyungpook Math. J. 18 (1978) 53–59. [10] S. Owa, H.M. Srivastava, Univalent and starlike generalized hypergeometric functions, Canad. J. Math. 39 (1987) 1057–1077. [11] S. Owa, M. Nunokawa, H. Saitoh, H.M. Srivastava, Close-to-convexity, starlikeness, and convexity of certain analytic functions, Appl. Math. Lett. 15 (1) (2002) 63–69. [12] S. Ruscheweyh, New criteria for univalent functions, Proc. Am. Math. Soc. 49 (1975) 109–115. [13] H.M. Srivastava, Some families of fractional derivative and other linear operators associated with analytic, univalent and multivalent functions, in: K.S. Lakshmi (Ed.), Proceedings of the International Conference on Analysis and Its Applications, Allied Publishers Limited, New Delhi and Chennai, 2001, pp. 209–243.