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Some convolution conditions for starlikeness and convexity of meromorphically multivalent functions
PERGAMON
Applied Mathematics Letters
Applied Mathematics Letters 16 (2003) 13-16
www.elsevier.com/locate/aml
S o m e Convolution C o n d i t i o n s for Starlikeness and C o n v e x i t y of M e r o m o r p h i c a l l y Multivalent Functions JIN-LIN LIU Department of Mathematics,YangzhouUniversity Yangzhou 225002, Jiangsu, P.R. China j lliucn@yahoo, com. cn H. M . SItIVASTAVA Department of Mathematics and Statistics,Universityof Victoria Victoria, BritishColumbia V 8 W 3P4, Canada harimsri~nath, uvic. ca
(Received December 2001; accepted February 2002) Abstract--By making use of the Hadamard product (or convolution), the authors derive necessary and sufficient conditions for a suitably normalized meromorphically p-valent function to be in the familiar classes of meromorphic p-valently starlike and meromorphic p-valently convex functions in the punctured unit disk. ~ 2002 Elsevier Science Ltd. All rights reserved. Keywords--Hadamard product (or convolution), Meromorphic functions, Multivalent functions, Analytic functions, Starlike functions, Convex functions.
1. I N T R O D U C T I O N
AND
DEFINITIONS
Let ~-~p denote the class of functions f normalized by c~
f (Z) = z -p + E a k
z k-p
(p e N := {1,2,3,...}),
(1.1)
k=l
which are analytic and p-valent in the punctured unit disk U* := {z: z e C and 0 < [z I < 1} = U\{O}. A function f E ~-~p is said to be in the class S~ (a) of meromorphic p-valently starlike functions of order a in U* if and only if Yt\ f(z) ] <-a
(zeU; 0=
(1.2)
The present investigation was supported, in part, by the Natural Sciences and Engineering Research Council of Canada under Grant OGP0007353. 0893-9659/02/$ - see front matter (~) 2002 Elsevier Science Ltd. All rights reserved. PII: S0893-9659(02)00138-6
Typeset by .4~iS-TEX
14
J.-L. Liu AND H. M. SRIVASTAVA
Furthermore, a function f E ~-~pis said to be in the class K:p (a) of meromorphic p-valently convex functions of order a in U* if and only if
zf"(z) ff¢ 1 +
(zeU; 0
f, (z) ] < - a
(1.3)
peN).
Throughout the present investigation, it should be understood that functions such as those occurring in (1.2) and (1.3), which have removable singularities at z = 0, have had these singularities removed in statements like (1.2) and (1.3). Clearly, we have f (z) e ~p (a) ¢==~ z f ' ( z ) e 5~ (a)
(0 <=a < p; p e N),
(1.4)
which obviously is analogous to the well-known Alexander equivalence (see, for details, [1]). For functions f E ~-~.pgiven by (1.1) and g E ~-~p given by cx)
g (z) = z -p + E
bk z k-p
(p E N),
(1.5)
k=l
we define the Hadamard product (or convolution) of f and g by oo
( f * g ) ( z ) : = z -p + E
(1.6)
ak bk z k-p = (g * f ) (z) .
k=l
Many important properties and characteristics of various interesting subclasses of the class ~-~v of meromorphically p-valent functions, including (for example) the classes £p (a) and ~v(a) defined above, were investigated extensively by (among others) Aouf et al. [2-4], Cho and Owa [5], Joshi and Srivastava [6], Kulkarni et al. [7], Liu and Srivastava [8], Mogra [9], Owa et al. [10], Srivastava et al. [11], Uralegaddi and Somanath [12], and Yang [13] (see also [14]). The present sequel to these earlier papers is motivated essentially by the work of Silverman et al. [15]. We aim here at deriving several characterizations of the classes 8~ (a) and K:p (a) by making use of the Hadamard product (or convolution) defined by (1.6).
2.
CONVOLUTION
CONDITION
FOR
THE CLASS $~(a)
We begin by proving the following general result. THEOREM 1. Let f E ~-~p. Then
f(z) ] < -a
(z E U ;
-~
pen
),
(2.1)
if and only if
f (z) * where :=
1 - ~z zp (1 - z) 2
# 0
1 + ~ + p (1 + e -2i~) - 2ae -ia p (1 + e -2ix) - 2ae - ~
(z E U*)
and
(2.2)
1~[ = 1.
(2.3)
PROOF. It is easily seen that condition (2.1) holds true if and only if e i~ [zf' ( z ) / f (z)] + a + i p s i n A # 1 -
pcosA - a
l+t¢
(z C U*; I~1 = 1; . # - 1 ) ,
(2.4)
Meromorphically Multivalent Functions
15
which, upon simplification, yields (1 + ~) zf' (z) -t- (pr~ -t- 2ae -iA - pe -2iA) f (z) # 0
(z • U*; Ix[ = 1).
(2.5)
Next, for f • )--~.pgiven by (1.1), we have 1
f(Z)*zp(l_z)2 and
(z • U*)
zf'(z)+(p+l)f(z)
1 f (z)* zp(1 - z) = f (z)
(2.6)
(z • U*).
(2.7)
Finally, in view of convolutions (2.6) and (2.7), (2.5) can be rewritten in its equivalent form
z,(1-
z) a
zP-(1--z)]
So
zP(
(z • U*; [h I = 1), that is, f (z) * 2ae-C~ - p (1 + e -2c~) + [1 q- a + p (1 -t- e -2'x) - 2ae -i~] z ~ 0 zP (1 - z) 2
(, • u*; I~l = 1), which leads us at once to the convolution condition (2.2). This evidently completes the proof of Theorem 1. By setting ), = 0 in Theorem 1, we immediately deduce the following characterization of the
class s; (~). COROLLARY 1.
Let f • Y~p. Then 1 -
f • 3p (0~) ¢=~ f (Z) * Zp
where 0 :=
(1
1 -t- ~-t- 2 ( p - a) 2 (p - c~)
3. C O N V O L U T I O N
CONDITION
Oz
z"2) # 0
and FOR
(z•U*),
(2.8)
l~l = 1. THE
(2.9) CLASS
/Cp(a)
In this section, we first prove another general result contained in the following. THEOREM 2. Let f E )"~p. Then 9l
{ ( e i~
1+
if(z) J
<-a
(
zeU;
-~
0
pen
)
,
(3.1)
if and only if f (z) *
p - [2 + p + ( p - 1 ) ~ ] z + ( p + 1)f~z 2 zp (1 - z ) 3 ¢ 0
(Z • U*),
(3.2)
where f~ and ~ are given (as also in Theorem 1) by (2.3). PROOF. In view of the Alexander-type equivalence (1.4), we find from Theorem 1 that condition (3.1) holds true if and only if
1-
zf'(Z)*zp(l_z)2
(
=f(z)*Z~kzp(l_z)2 ) 50
(z•U*),
which readily yields the desired assertion (3.2) of Theorem 2. In its special case, then A = 0, we obtain the following characterization of the class K:p(a).
16
J.-L. LIu AND H. M. SRIVASTAVA
COROLLARY 2. L e t f E ~-:~p. T h e n
/ e / c , (~) .=~ / (z) •
p-
[2 + p +
(p-
1) O ] z + (p + 1) O z 2
zp (1 - z) 3
0
(z E U * ) ,
(3.3)
where e and ~ are given (as also in Corollary 1) by (2.9).
REFERENCES 1. P.L. Duren, Univalent functions, In Grundlehren der Mathematischen Wissenscha/ten, Bd., Volume P59, Springer-Verlag, New York, (1983). 2. M.K. Aouf, New criteria for multivalent meromorphic starlike functions of order alpha, Proc. Japan Acad. Ser. A Math. Sci. 69, 66-70, (1993). 3. M.K. Aouf and H.M. Hossen, New criteria for meromorphic p-valent starlike functions, Tsukuba J. Math. 17, 481-486, (1993). 4. M.K. Aouf and H.M. Srivastava, A new criterion for meromorphically p-valent convex functions of order alpha, MaSh. Sci. Res. Hot-Line 1 (8), 7-12, (1997). 5. N.E. Cho and S. Owa, On certain classes of meromorphically p-valent starlike functions, In New Developments in Univalent Function Theory, Volume 8P1, (Edited by S. Owa), pp. 159-165, Kyoto; August 4-7, 1992, Sfirikaisekikenkyfisho KSkyfiroku, Research Institute for Mathematical Sciences, Kyoto University, Kyoto, (1993). 6. S.B. Joshi and H.M. Srivastava, A certain family of meromorphically multivalent functions, Computers Math. Appl. 38 (3/4), 201-211, (1999). 7. S.R. Kulkarni, U.H. Naik and H.M. Srivastava, A certain class of meromorphically p-valent quasi-convex functions, Pan Amer. Math. J. 8 (1), 5764, (1998). 8. J.-L. Liu and H.M. Srivnstava, A linear operator and associated families of meromorphically multivalent functions, J. M a ~ . Anal. Appl. 259, 566-581, (2001). 9. M.L. Mogra, Meromorphic multivalent functions with positive coefficients I and II, Math. Japon. 35, 1-11 and 1089-1098, (1990). 10. S. Owa, H.E. Darwish and M.K. Aouf, Meromorphic multivalent functions with positive and fixed second coefficients, Math. Japon. 46, 231-236, (1997). 11. H.M. Srivastava, H.M. Hossen and M.K. Aouf, A unified presentation of some classes of meromorphically multivalent functions, Computers Math. Appl. 38 (11/1.2), 63-70, (1999). 12. B.A. Uralegaddi and C. Somanath, Certain classes of meromorphic multivalent functions, Tamkang J. Math. 23, 223-231, (1992). 13. D.-C. Yang, On new subclasses of meromorphic p-valent functions, J. Math. Res. Exposition 15, 7-13, (1995). 14. H.M. Srivastava and S. Owa, Editors, Current Topics in Analytic Function Theory, World Scientific, Singapore, (1992). 15. H. Silverman, E.M. Silvia and D. Telage, Convolution conditions for convexity, starlikeness and spiral-likeness, Math. Zeitschr. 162, 125-130, (1978).
Applied Mathematics Letters
Applied Mathematics Letters 16 (2003) 13-16
www.elsevier.com/locate/aml
S o m e Convolution C o n d i t i o n s for Starlikeness and C o n v e x i t y of M e r o m o r p h i c a l l y Multivalent Functions JIN-LIN LIU Department of Mathematics,YangzhouUniversity Yangzhou 225002, Jiangsu, P.R. China j lliucn@yahoo, com. cn H. M . SItIVASTAVA Department of Mathematics and Statistics,Universityof Victoria Victoria, BritishColumbia V 8 W 3P4, Canada harimsri~nath, uvic. ca
(Received December 2001; accepted February 2002) Abstract--By making use of the Hadamard product (or convolution), the authors derive necessary and sufficient conditions for a suitably normalized meromorphically p-valent function to be in the familiar classes of meromorphic p-valently starlike and meromorphic p-valently convex functions in the punctured unit disk. ~ 2002 Elsevier Science Ltd. All rights reserved. Keywords--Hadamard product (or convolution), Meromorphic functions, Multivalent functions, Analytic functions, Starlike functions, Convex functions.
1. I N T R O D U C T I O N
AND
DEFINITIONS
Let ~-~p denote the class of functions f normalized by c~
f (Z) = z -p + E a k
z k-p
(p e N := {1,2,3,...}),
(1.1)
k=l
which are analytic and p-valent in the punctured unit disk U* := {z: z e C and 0 < [z I < 1} = U\{O}. A function f E ~-~p is said to be in the class S~ (a) of meromorphic p-valently starlike functions of order a in U* if and only if Yt\ f(z) ] <-a
(zeU; 0=
(1.2)
The present investigation was supported, in part, by the Natural Sciences and Engineering Research Council of Canada under Grant OGP0007353. 0893-9659/02/$ - see front matter (~) 2002 Elsevier Science Ltd. All rights reserved. PII: S0893-9659(02)00138-6
Typeset by .4~iS-TEX
14
J.-L. Liu AND H. M. SRIVASTAVA
Furthermore, a function f E ~-~pis said to be in the class K:p (a) of meromorphic p-valently convex functions of order a in U* if and only if
zf"(z) ff¢ 1 +
(zeU; 0
f, (z) ] < - a
(1.3)
peN).
Throughout the present investigation, it should be understood that functions such as those occurring in (1.2) and (1.3), which have removable singularities at z = 0, have had these singularities removed in statements like (1.2) and (1.3). Clearly, we have f (z) e ~p (a) ¢==~ z f ' ( z ) e 5~ (a)
(0 <=a < p; p e N),
(1.4)
which obviously is analogous to the well-known Alexander equivalence (see, for details, [1]). For functions f E ~-~.pgiven by (1.1) and g E ~-~p given by cx)
g (z) = z -p + E
bk z k-p
(p E N),
(1.5)
k=l
we define the Hadamard product (or convolution) of f and g by oo
( f * g ) ( z ) : = z -p + E
(1.6)
ak bk z k-p = (g * f ) (z) .
k=l
Many important properties and characteristics of various interesting subclasses of the class ~-~v of meromorphically p-valent functions, including (for example) the classes £p (a) and ~v(a) defined above, were investigated extensively by (among others) Aouf et al. [2-4], Cho and Owa [5], Joshi and Srivastava [6], Kulkarni et al. [7], Liu and Srivastava [8], Mogra [9], Owa et al. [10], Srivastava et al. [11], Uralegaddi and Somanath [12], and Yang [13] (see also [14]). The present sequel to these earlier papers is motivated essentially by the work of Silverman et al. [15]. We aim here at deriving several characterizations of the classes 8~ (a) and K:p (a) by making use of the Hadamard product (or convolution) defined by (1.6).
2.
CONVOLUTION
CONDITION
FOR
THE CLASS $~(a)
We begin by proving the following general result. THEOREM 1. Let f E ~-~p. Then
f(z) ] < -a
(z E U ;
-~
pen
),
(2.1)
if and only if
f (z) * where :=
1 - ~z zp (1 - z) 2
# 0
1 + ~ + p (1 + e -2i~) - 2ae -ia p (1 + e -2ix) - 2ae - ~
(z E U*)
and
(2.2)
1~[ = 1.
(2.3)
PROOF. It is easily seen that condition (2.1) holds true if and only if e i~ [zf' ( z ) / f (z)] + a + i p s i n A # 1 -
pcosA - a
l+t¢
(z C U*; I~1 = 1; . # - 1 ) ,
(2.4)
Meromorphically Multivalent Functions
15
which, upon simplification, yields (1 + ~) zf' (z) -t- (pr~ -t- 2ae -iA - pe -2iA) f (z) # 0
(z • U*; Ix[ = 1).
(2.5)
Next, for f • )--~.pgiven by (1.1), we have 1
f(Z)*zp(l_z)2 and
(z • U*)
zf'(z)+(p+l)f(z)
1 f (z)* zp(1 - z) = f (z)
(2.6)
(z • U*).
(2.7)
Finally, in view of convolutions (2.6) and (2.7), (2.5) can be rewritten in its equivalent form
z,(1-
z) a
zP-(1--z)]
So
zP(
(z • U*; [h I = 1), that is, f (z) * 2ae-C~ - p (1 + e -2c~) + [1 q- a + p (1 -t- e -2'x) - 2ae -i~] z ~ 0 zP (1 - z) 2
(, • u*; I~l = 1), which leads us at once to the convolution condition (2.2). This evidently completes the proof of Theorem 1. By setting ), = 0 in Theorem 1, we immediately deduce the following characterization of the
class s; (~). COROLLARY 1.
Let f • Y~p. Then 1 -
f • 3p (0~) ¢=~ f (Z) * Zp
where 0 :=
(1
1 -t- ~-t- 2 ( p - a) 2 (p - c~)
3. C O N V O L U T I O N
CONDITION
Oz
z"2) # 0
and FOR
(z•U*),
(2.8)
l~l = 1. THE
(2.9) CLASS
/Cp(a)
In this section, we first prove another general result contained in the following. THEOREM 2. Let f E )"~p. Then 9l
{ ( e i~
1+
if(z) J
<-a
(
zeU;
-~
0
pen
)
,
(3.1)
if and only if f (z) *
p - [2 + p + ( p - 1 ) ~ ] z + ( p + 1)f~z 2 zp (1 - z ) 3 ¢ 0
(Z • U*),
(3.2)
where f~ and ~ are given (as also in Theorem 1) by (2.3). PROOF. In view of the Alexander-type equivalence (1.4), we find from Theorem 1 that condition (3.1) holds true if and only if
1-
zf'(Z)*zp(l_z)2
(
=f(z)*Z~kzp(l_z)2 ) 50
(z•U*),
which readily yields the desired assertion (3.2) of Theorem 2. In its special case, then A = 0, we obtain the following characterization of the class K:p(a).
16
J.-L. LIu AND H. M. SRIVASTAVA
COROLLARY 2. L e t f E ~-:~p. T h e n
/ e / c , (~) .=~ / (z) •
p-
[2 + p +
(p-
1) O ] z + (p + 1) O z 2
zp (1 - z) 3
0
(z E U * ) ,
(3.3)
where e and ~ are given (as also in Corollary 1) by (2.9).
REFERENCES 1. P.L. Duren, Univalent functions, In Grundlehren der Mathematischen Wissenscha/ten, Bd., Volume P59, Springer-Verlag, New York, (1983). 2. M.K. Aouf, New criteria for multivalent meromorphic starlike functions of order alpha, Proc. Japan Acad. Ser. A Math. Sci. 69, 66-70, (1993). 3. M.K. Aouf and H.M. Hossen, New criteria for meromorphic p-valent starlike functions, Tsukuba J. Math. 17, 481-486, (1993). 4. M.K. Aouf and H.M. Srivastava, A new criterion for meromorphically p-valent convex functions of order alpha, MaSh. Sci. Res. Hot-Line 1 (8), 7-12, (1997). 5. N.E. Cho and S. Owa, On certain classes of meromorphically p-valent starlike functions, In New Developments in Univalent Function Theory, Volume 8P1, (Edited by S. Owa), pp. 159-165, Kyoto; August 4-7, 1992, Sfirikaisekikenkyfisho KSkyfiroku, Research Institute for Mathematical Sciences, Kyoto University, Kyoto, (1993). 6. S.B. Joshi and H.M. Srivastava, A certain family of meromorphically multivalent functions, Computers Math. Appl. 38 (3/4), 201-211, (1999). 7. S.R. Kulkarni, U.H. Naik and H.M. Srivastava, A certain class of meromorphically p-valent quasi-convex functions, Pan Amer. Math. J. 8 (1), 5764, (1998). 8. J.-L. Liu and H.M. Srivnstava, A linear operator and associated families of meromorphically multivalent functions, J. M a ~ . Anal. Appl. 259, 566-581, (2001). 9. M.L. Mogra, Meromorphic multivalent functions with positive coefficients I and II, Math. Japon. 35, 1-11 and 1089-1098, (1990). 10. S. Owa, H.E. Darwish and M.K. Aouf, Meromorphic multivalent functions with positive and fixed second coefficients, Math. Japon. 46, 231-236, (1997). 11. H.M. Srivastava, H.M. Hossen and M.K. Aouf, A unified presentation of some classes of meromorphically multivalent functions, Computers Math. Appl. 38 (11/1.2), 63-70, (1999). 12. B.A. Uralegaddi and C. Somanath, Certain classes of meromorphic multivalent functions, Tamkang J. Math. 23, 223-231, (1992). 13. D.-C. Yang, On new subclasses of meromorphic p-valent functions, J. Math. Res. Exposition 15, 7-13, (1995). 14. H.M. Srivastava and S. Owa, Editors, Current Topics in Analytic Function Theory, World Scientific, Singapore, (1992). 15. H. Silverman, E.M. Silvia and D. Telage, Convolution conditions for convexity, starlikeness and spiral-likeness, Math. Zeitschr. 162, 125-130, (1978).