Inclusion and neighborhood properties for certain classes of multivalently analytic functions of complex order associated with the convolution structure

Applied Mathematics and Computation 212 (2009) 66–71

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Inclusion and neighborhood properties for certain classes of multivalently analytic functions of complex order associated with the convolution structure H.M. Srivastava a,*, Sevtap Sümer Eker b, Bilal Sß eker b a b

Department of Mathematics and Statistics, University of Victoria, Victoria, British Columbia V8W 3R4, Canada Department of Mathematics, Faculty of Science and Letters, Dicle University, TR-21280 Diyarbakır, Turkey

a r t i c l e

i n f o

a b s t r a c t

Keywords: Analytic functions Univalent functions Multivalently analytic functions Hadamard product (or convolution) Coefficient inequalities Inclusion properties Neighborhood properties Ruscheweyh derivative operator

Making use of the familiar convolution structure of analytic functions, in this paper we introduce and investigate two new subclasses of multivalently analytic functions of complex order. Among the various results obtained here for each of these function classes, we derive the coefficient inequalities and other interesting properties and characteristics for functions belonging to the class introduced here. Ó 2009 Elsevier Inc. All rights reserved.

1. Introduction and definitions Let Ap ðkÞ denote the class of functions of the form:

f ðzÞ ¼ zp þ

1 X

aj zj

ðp < k; k; p 2 N; N :¼ f1; 2; 3; . . .gÞ;

ð1:1Þ

j¼k

which are analytic and p-valent in the open unit disk

U ¼ fz : z 2 C and jzj < 1g: For the functions f ; g 2 Ap ðkÞ, where f is given by (1.1) and g is given by

gðzÞ ¼ zp þ

1 X

bj zj

ðp < k; k; p 2 NÞ;

j¼k

the Hadamard product (or convolution) f  g is defined (as usual) by

ðf  gÞðzÞ :¼ zp þ

1 X

aj bj zj ¼: ðg  f ÞðzÞ ðz 2 UÞ:

j¼k

We denote by Tp ðkÞ the subclass of Ap ðkÞ consisting of functions of the form:

* Corresponding author. E-mail addresses: [email protected] (H.M. Srivastava), [email protected] (S.S. Eker), [email protected] (B. Sßeker). 0096-3003/$ - see front matter Ó 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2009.01.077

ð1:2Þ

H.M. Srivastava et al. / Applied Mathematics and Computation 212 (2009) 66–71 1 X

f ðzÞ ¼ zp 

a j zj



 p < k; aj = 0 ðj = kÞ; k; p 2 N ;

67

ð1:3Þ

j¼k

which are p-valent in U (see, for details [4,11]). For a given function g 2 Ap ðkÞ defined by

gðzÞ ¼ zp þ

1 X

bj zj



 p < k; bj = 0 ðj = kÞ; k; p 2 N ;

ð1:4Þ

j¼k

we introduce here a new class Sg ðp; k; b; m; nÞ of functions belonging to the subclass of Tp ðkÞ, which consists of functions f ðzÞ of the form (1.3) satisfying the following inequality:

 !  1 zn ðf  gÞðmþnÞ ðzÞ    ðp  mÞ  n  < 1 ðmÞ  b ðf  gÞ ðzÞ

ð1:5Þ

ðm þ n < p; n; p 2 N; m 2 N0 :¼ N [ f0g; b 2 C n f0g; z 2 UÞ; where ðjÞn denotes the falling factorial defined as follows:

 

j

ðjÞ0 ¼ 1 ¼:

0

and ðjÞn ¼ jðj  1Þ    ðj  n þ 1Þ ¼: n!

 

j n

ðn 2 NÞ:

We note that there are several interesting new or known subclasses of our function class Sg ðp; k; b; m; nÞ. For example, if we set

m ¼ 0; n ¼ 1 and b ¼ pð1  aÞ ðp 2 N; 05a < 1Þ in (1.5), then Sg ðp; k; b; m; nÞ reduces to the function class studied recently by Ali et al. [1]. On the other hand, when n ¼ 1, if the coefficients bj in (1.4) are chosen as follows:

 bj ¼

kþj1



jp

ðk > p; p 2 NÞ

and k is replaced by k þ p in (1.2) and (1.3), then we obtain the function class introduced and investigated earlier by Raina and Srivastava [7], which involves the familiar Ruscheweyh derivative operator. Furthermore, if we choose n ¼ 1 in (1.5), we obtain the function class Sg ðp; n; b; mÞ which was studied by Prajabat et al. [6] (see also [3,9,12], and the references cited in each of them). For a given function gðzÞ defined by

gðzÞ ¼ zp þ

1 X

bj zj 2 Ap ðkÞ



 p < k; bj > 0 ðj = kÞ; k; p 2 N ;

j¼k

let Pg ðp; k; b; m; n; lÞ denote the subclass of Tp ðkÞ consisting of functions f ðzÞ of the form (1.3), which satisfy the following inequality:

 !  ðmÞ  1 ðf  gÞðzÞ   ðmþnÞ þ l ðf  gÞ ðzÞ  ðp  mÞ ðpÞn ð1  lÞ  n  < ðp  mÞn n  b z ðm þ n < p < k; n; p 2 N; m 2 N0 ;

ð1:6Þ

l = 0; b 2 C n f0g; z 2 UÞ:

Following recent investigations by several authors (see, for example [2,5,12], and others), if

f ðzÞ 2 Tp ðkÞ and d = 0; then we define the ðq; dÞ-neighborhood of the function f ðzÞ by

( Nqk;d ðf Þ ¼

h : h 2 Tp ðkÞ; hðzÞ ¼ zp 

1 X

cj zj and

j¼k

1 X

) j

qþ1

jaj  cj j 5 d :

ð1:7Þ

j¼k

It follows from the definition (1.7) that, if the identity function eðzÞ is given by

eðzÞ ¼ zp

ðp 2 N; z 2 UÞ;

ð1:8Þ

then

( Nqk;d ðeÞ ¼

h : h 2 Tp ðkÞ; hðzÞ ¼ zp 

1 X j¼k

We observe that

N02;d ðf Þ ¼ Nd ðf Þ

cj zj and

1 X j¼k

) j

qþ1

jcj j 5 d :

ð1:9Þ

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H.M. Srivastava et al. / Applied Mathematics and Computation 212 (2009) 66–71

and

N12;d ðf Þ ¼ Md ðf Þ; where Nd ðf Þ and Md ðf Þ denote, respectively, the d-neighborhoods of the function

f ðzÞ ¼ z 

1 X

aj zj



aj = 0 ðj = 2Þ



ð1:10Þ

j¼2

as defined by Ruscheweyh [8] and Silverman [10]. The object of the present paper is to investigate the various properties and characteristics of functions belonging to the above-defined classes

Sg ðp; k; b; m; nÞ and Pg ðp; k; b; m; n; lÞ: Apart from deriving coefficient bounds and coefficient inequalities for each of these classes, we establish several inclusion relationships involving the ðq; dÞ-neighborhoods of functions belonging to the general classes which are introduced above. 2. Coefficient bounds and coefficient inequalities We begin by proving a necessary and sufficient condition for the function f ðzÞ 2 Tp ðkÞ to be in each of the classes

Sg ðp; k; b; m; nÞ and Pg ðp; k; b; m; n; lÞ:

Theorem 1. Let f ðzÞ 2 Tp ðkÞ be given by (1.3). Then f ðzÞ is in the class Sg ðp; k; b; m; nÞ if and only if 1 X ðjÞm ½ðj  mÞn  ðp  mÞn þ jbjaj bj 5 jbjðpÞm

ð2:1Þ

j¼k

ðm þ n < p < k; m 2 N0 ; n 2 N; b 2 C n f0gÞ: Proof. Assume that f 2 Sg ðp; k; b; m; nÞ. Then, in view of (1.3)–(1.5), we have

zn ðf  gÞðmþnÞ ðzÞ  ðp  mÞn ðf  gÞðmÞ ðzÞ

R

!

ðf  gÞðmÞ ðzÞ

> jbj ðz 2 UÞ;

which yields



R

P1

 mÞn  ðp  mÞn aj bj zjp P jp ðpÞm  1 j¼k ðjÞm aj bj z

j¼k ðjÞm ½ðj

! > jbj ðz 2 UÞ:

ð2:2Þ

Putting z ¼ r ð0 5 r < 1Þ in (2.2), the expression in the denominator on the left-hand side of (2.2) remains positive for r ¼ 0 and also for all r 2 ð0; 1Þ. Hence, by letting r ! 1, the inequality (2.2) leads us to the desired assertion (2.1) of Theorem 1. Conversely, by applying the hypothesis (2.1) and setting j z j¼ 1, we find that

 P1   jp    zn ðf  gÞðmþnÞ ðzÞ   j¼k ðjÞm ½ðj  mÞ   n  ðp  mÞn aj bj z P  ðp  mÞn  ¼    jp     ðf  gÞðmÞ ðzÞ ðpÞm  1 j¼k ðjÞm aj bj z P1 j¼k ðjÞm ½ðj  mÞn  ðp  mÞn aj bj P 5 ðpÞm  1 j¼k ðjÞm aj bj h i P1 jbj ðpÞm  j¼k ðjÞm aj bj P < ¼ jbj: ðpÞm  1 j¼k ðjÞm aj bj Hence, by the maximum modulus principle, we infer that f 2 Sg ðp; k; b; m; nÞ, which completes the proof of Theorem 1. h In a similar manner, we can prove Theorem 2 below. Theorem 2. Let f ðzÞ 2 Tp ðkÞ be given by (1.3). Then f ðzÞ is in the class Pg ðp; k; b; m; n; lÞ if and only if 1 X ðj  nÞm ½lðjÞn  ðl  1ÞðpÞn aj bj 5 ðp  mÞn ½jbj  1 þ ðpÞm : j¼k

ðm þ n < p < k; m 2 N0 ; n 2 N; b 2 C n f0gÞ:

ð2:3Þ

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H.M. Srivastava et al. / Applied Mathematics and Computation 212 (2009) 66–71

3. A set of inclusion relationships We now establish some inclusion relationships for each of the function classes

Sg ðp; k; b; m; nÞ and Pg ðp; k; b; m; n; lÞ involving the ðq; dÞ-neighborhood defined by (1.7). Theorem 3. If bj = bk ðj = kÞ and

kjbjðpÞm ðkÞm ½ðk  mÞn  ðp  mÞn þ jbjbk

d¼

ðp > jbjÞ;

ð3:1Þ

then

Sg ðp; k; b; m; nÞ  N0k;d ðeÞ:

ð3:2Þ

Proof. Let f ðzÞ 2 Sg ðp; k; b; m; nÞ. Then, in view of the assertion (2.1) and the given condition that bj = bk ðj = kÞ, we get

ðkÞm ½ðk  mÞn  ðp  mÞn þ jbjbk

1 X

aj 5

j¼k

1 X ½ðj  mÞn  ðp  mÞn þ jbjðjÞm aj bj < jbjðpÞm ; j¼k

which implies that 1 X

aj 5

j¼k

jbjðpÞm : ðkÞm ½ðk  mÞn  ðp  mÞn þ jbjbk

ð3:3Þ

Furthermore, by rewriting the assertion (2.1) as follows: 1 X ðj  1Þ! ½ðj  mÞn  ðp  mÞn þ jbjjaj bj < jbjðpÞm ; ðj  mÞ! j¼k

we obtain 1 X j¼k

jaj 5

jbjðk  mÞ!ðpÞm ¼ d ðp > jbjÞ; ðk  1Þ!½ðk  mÞn  ðp  mÞn þ jbjbk

ð3:4Þ

which, by virtue of (1.9), establishes the inclusion relationship (3.2). h In an analogous manner, by applying the assertion (2.3), instead of the assertion (2.1), to the functions in the class Pg ðp; k; b; m; n; lÞ, we can prove the following inclusion relationship. Theorem 4. If bj = bk ðj = kÞ and

d¼

kðp  mÞn ½jbj  1 þ ðpÞm ðk  m  nÞ! ðk  nÞ!½lðkÞn  ðl  1ÞðpÞn bk

ðl > 1Þ;

then

Pg ðp; k; b; m; n; lÞ  N0k;d ðeÞ: 4. Neighborhood properties In this concluding section, we determine the neighborhood properties for each of the function classes

SgðaÞ ðp; k; b; m; nÞ and PgðaÞ ðp; k; b; m; n; lÞ; which are defined as follows. ða Þ

Definition 1. A function f ðzÞ 2 Tp ðkÞ is said to be in the class Sg ðp; k; b; m; nÞ if there exists a function hðzÞ 2 Sg ðp; k; b; m; nÞ such that

   f ðzÞ    < p  a ðz 2 U; 0 5 a < pÞ:  1 hðzÞ 

ð4:1Þ

Definition 2. A function f ðzÞ 2 Tp ðkÞ is said to be in the class PðgaÞ ðp; k; b; m; n; lÞ if there exists a function hðzÞ 2 Pg ðp; k; b; m; n; lÞ such that the inequality (4.1) holds true.

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Theorem 5. If hðzÞ 2 Sg ðp; k; b; m; nÞ and

a¼p

d ðkÞm ½ðk  mÞn  ðp  mÞn þ jbjbk  ½ðk  mÞn  ðp  mÞn þ jbjbk  jbjpm ; ðkÞm k qþ1

ð4:2Þ

then

Nqk;d ðhÞ  SðgaÞ ðp; k; b; m; nÞ: Proof. Suppose that f ðzÞ 2 Nqk;d ðhÞ. We then find from (1.7) that 1 X

j

  aj  cj  5 d;

qþ1 

j¼k

which readily implies that 1 X

jaj  cj j 5

j¼k

d k

qþ1

ðk 2 NÞ:

Next, since hðzÞ 2 Sg ðp; k; b; m; nÞ, we find from (3.3) that 1 X

cj 5

j¼k

jbjðpÞm ; ðkÞm ½ðk  mÞn  ðp  mÞn þ jbjbk

so that

 P1    f ðzÞ ja  c j 5 j¼k Pj 1 j 5 d ðkÞm ½ðk  mÞn  ðp  mÞn þ jbjbk   1  1 hðzÞ qþ1 ðkÞm k j¼k c j  ½ðk  mÞn  ðp  mÞn þ jbjbk  jbjðpÞm ¼ p  a ðz 2 U; 0 5 a < pÞ; where a is given, as before, by (4.2). Thus, by Definition 1, f 2 SðgaÞ ðp; k; b; m; nÞ for a given by (4.2). This evidently proves Theorem 5. h The proof of Theorem 6 below (based upon Definition 2) is similar to that of Theorem 5 above. We, therefore, omit the details involved. Theorem 6. If hðzÞ 2 Pg ðp; k; b; m; n; lÞ and

a¼p

d qþ1

k

ðk  nÞm ½lðkÞn  ðl  1ÞðpÞn bk ; ðk  nÞm ½lðkÞn  ðl  1ÞðpÞn   ðp  mÞn ½jbj  1 þ ðpÞm 

ð4:3Þ

then

Nqk;d ðhÞ  PgðaÞ ðp; k; b; m; n; lÞ: Acknowledgements The present investigation was supported, in part, by the Natural Sciences and Engineering Research Council of Canada under Grant OGP0007353. References [1] R.M. Ali, M.H. Hussain, V. Ravichandran, K.G. Subramanian, A class of multivalent functions with negative coefficients defined by convolution, Bull. Korean Math. Soc. 43 (2006) 179–188. [2] O. Altintasß, H. Irmak, H.M. Srivastava, Neighborhoods for certain subclasses of multivalently analytic functions defined by using a differential operator, Comput. Math. Appl. 55 (2008) 331–338. [3] O. Altintasß, H. Irmak, S. Owa, H.M. Srivastava, Coefficient bounds for some families of starlike and convex functions of complex order, Appl. Math. Lett. 20 (2007) 1218–1222. [4] P.L. Duren, Univalent Functions, Grundlehren der Mathematischen Wissenschaften, Band 259, Springer-Verlag, New York, Berlin, Heidelberg and Tokyo, 1983. [5] B.A. Frasin, M. Darus, Integral means and neighborhoods for analytic univalent functions with negative coefficients, Soochow J. Math. 30 (2004) 217– 223. [6] J.K. Prajapat, R.K. Raina, H.M. Srivastava, Inclusion and neighborhood properties for certain classes of multivalently analytic functions associated with the convolution structure, J. Inequal. Pure Appl. Math. 8 (1) (2007). Article 7, 1–8 (electronic). [7] R.K. Raina, H.M. Srivastava, Inclusion and neighborhood properties of some analytic and multivalent functions, J. Inequal. Pure Appl. Math. 7 (1) (2006). Article 5, 1–6 (electronic). [8] S. Ruscheweyh, Neighborhoods of univalent functions, Proc. Amer. Math. Soc. 81 (1981) 521–527. [9] B. Shrutha Keerthi, A. Gangadharan, H.M. Srivastava, Neighborhoods of certain subclasses of analytic functions of complex order with negative coefficients, Math. Comput. Modelling 47 (2008) 271–277.

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[10] H. Silverman, Neighborhoods of classes of analytic functions, Far East J. Math. Sci. 3 (1995) 165–169. [11] H.M. Srivastava, S. Owa (Eds.), Current Topics in Analytic Function Theory, World Scientific Publishing Company, Singapore, New Jersey, London and Hong Kong, 1992. [12] H.M. Srivastava, K. Suchithra, B.A. Stephen, S. Sivasubramanian, Inclusion and neighborhood properties of certain subclasses of analytic and multivalent functions of complex order, J. Inequal. Pure Appl. Math. 7 (5) (2006). Article 191, 1–8 (electronic).