NEW SUBCLASS OF ANALYTIC FUNCTIONS IN CONICAL DOMAIN ASSOCIATED WITH A LINEAR OPERATOR∗

Muhammed ARIF　Shahid MAHMOODDepartment of Mathematics，Abdulwali Khan University Mardan，PakistanE-mail:marifmaths@yahoo.com;shahidmahmood757@gmail.comJanusz SOK´O LDepartment of Mathematics，Rzesz´ow University of Technology，Al.Powsta´nc´ow Warszawy 12，35-959 Rzesz´ow，PolandE-mail:jsokol@prz.edu.pl

Jacek DZIOKInstitute of Mathematics，University of Rzesz´ow，ul.Rejtana 16A，35-310 Rzesz´ow，PolandE-mail:jdziok@univ.rzeszow.pl

NEW SUBCLASS OF ANALYTIC FUNCTIONS IN CONICAL DOMAIN ASSOCIATED WITH A LINEAR OPERATOR∗

Muhammed ARIFShahid MAHMOOD
Department of Mathematics，Abdulwali Khan University Mardan，Pakistan
E-mail:marifmaths@yahoo.com;shahidmahmood757@gmail.com
Janusz SOK´O L†
Department of Mathematics，Rzesz´ow University of Technology，Al.Powsta´nc´ow Warszawy 12，35-959 Rzesz´ow，Poland
E-mail:jsokol@prz.edu.pl

Jacek DZIOK
Institute of Mathematics，University of Rzesz´ow，ul.Rejtana 16A，35-310 Rzesz´ow，Poland
E-mail:jdziok@univ.rzeszow.pl

The main object of the present paper is to investigate a number of useful properties such as inclusion relations，distortion bounds，coefficient estimates，subordination results，the Fekete-Szeg¨o problem and some other for a new subclass of analytic functions，which are defined here by means of linear operator.Relevant connections of the results presented here with those obtained in earlier works are also pointed out.

analytic functions;subordination;functions with positive real part;linear operators;conic domains

2010 MR Subject Classification30C45;30C50

1　Introduction and Definitions

Let A be the class of functions having form

which are analytic in the open unit diskFurther，we denote the class S of all functions in A which are univalent indenote the classes of starlike，convex，close-to-convex and quasi-convex of order β，respectively.For details see［1］. Goodman［2］introduced the class UCV of uniformly convex functions.A functionis in the class UCV if for every circular arc ξ⊂E，with center in E，the arc f（ξ）is convex.An interesting characterization of class UCV was given in［3］，see also［4］，as

In［5］，see also［6］，it was introduced the class k-uniformly convex functions，k≥0，denoted by k-UCV and the class k-ST related to k-UCV by Alexandar type relation，i.e.，f（z）∈

In［5］and［6］，respectively，their geometric definitions and connections with the conic domains were also considered.If k≥0，then the class k-UCV is defined purely geometrically as a subclass of univalent functions which map the intersection of E with any disk centered atk，onto a convex domain.Therefore，the notion of k-uniform convexity is a generalization of the notion of convexity.Observe that，if k=0 then the center ζ is the origin and the class k-UCV reduces to the class C.Moreover for k=1 it coincides with the class of uniformly convex functions UCV introduced by Goodman［2］and studied extensively by Rønning［4］and independently by Ma and Minda［3］.We note that the class k-UCV started much earlier in papers［7，8］with some additional conditions but without the geometric interpretation.

A lot of the author’s investigated the properties of the classand their generalizations in several directions e.g.see［4，6，9-12］.

For two analytic functions

the convolution（Hadamard product）of is defined as

Now，we recall the incomplete beta functiondefined by

Using the technique of convolution，Carlson and Shaffer［14］defined a linear operator

From（1.4）the following identity can easily be derived

Throughout this paper，we assume thatunless otherwise stated.

Definition 1.1（see［16］）A functionis in the classif and only if

The boundary∂Ωkof the above set becomes the imaginary axis when k=0，a hyperbola when 0＜k＜1，a parabola when k=1，and an ellipse when k＞1.All of these curves have the vertex at the point k/（k+1）.Therefore the domain Ωk，γis elliptic for k＞1，hyperbolic when，parabolic for k=1 and right half plane when k=0;ever symmetric with respect to the real axis.

The function which plays the role of extremal functions for these conic regions are given as

In［18］was considered a related class，where h is a convex univalent function analytic in E withThereforebecomes the classdefined in［19］.In this paper we consider a special case of the classNamely we put

in（1.7）and in（1.8）.Then we obtain a class defined below.

Applying（1.6），we can write（1.12）as

By taking specific values to parameters，we obtain many important subclasses studied by various authors in earlier papers（see for details［19-25］）and list some of them as follows:

From the above special cases，we note that this class provides a continuous passage from the class of starlike functions to the class of convex functions.

2　Preliminary Results

Lemma 2.1For a positive integer t，we have

belongs to the class C（0）of convex functions.

3　Main Results

This proves（3.1）.Noting that the univalent functionmaps the diskonto a region which is convex and symmetric with respect to the real axis，we see

Using（3.4）and（3.5），we obtain

for z∈E.Consequently，subordination（3.4）leads us to

This completes the proof.

ProofLet

Now comparing the coefficients of zn，we obtain

To prove（3.7）we apply mathematical induction.So forwe have from（3.11）

which shows that（3.7）holds for，we have from（3.11）

which shows that（3.7）holds fo r.Let us assume that（3.7）is true forthat is，

Using（3.11）and（3.13），we have

Applying（2.1），we have

Consequently，using mathematical induction，we have proved that（3.7）holds true for all n，n≥2.This completes the proof.

Taking a=δ+1 and c=1 in Theorem 3.2，we obtain the following corollary.

Since f1（z）is univalent，so

Now using Theorem 3.2，we have

and hence

Using（3.19）in（3.18），we obtain

For any complex number ν，we have

Then（3.20）can be written as

where ν is defined by（3.17）.Now，taking absolute value on both sides and using Lemma 2.4，we obtain the required result.

Taking b=1，2，a=1 and c=1 in Theorem 3.7 we obtain the following corollary studied by Srivastava et al.［30］.

Corollary 3.10Let 0≤k＜∞be fixed and letThen for a complex number υ，

where ν2is given by（3.23）andare given by（2.2）.

ProofLet we note that

because from（3.24）it follows that

because of（3.24）.

By taking a=1，c=1 and b=1，2 in Theorem 3.11，we obtain the following results.

Corollary 3.12（see［25］） If a functionhas form（1.1）and it satisfies

Corollary 3.13（see［27］）If a functionhas form（1.1）and it satisfies

Corollary 3.14（see［33］）If a functionhas form（1.1）and it satisfies

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∗April 3，2015.

†Janusz SOK´O L.