Table of Contents
ISRN Mathematical Analysis
Volumeย 2012, Article IDย 698307, 11 pages
http://dx.doi.org/10.5402/2012/698307
Research Article

On Certain Subclasses of Meromorphic Functions with Positive and Fixed Second Coefficients Involving the Liu-Srivastava Linear Operator

1Research and P.G. Studies in Mathematics, Governt Arts College (Men), Tamilnadu Krishnagiri 635001, India
2P.G. Studies in Mathematics, Government Science College, Karnataka Chitradurga 577501, India
3Department of Mathematics, Adhiyamaan College of Engineering (Autonomous), Tamilnadu Hosur 635109, India

Received 23 December 2011; Accepted 16 February 2012

Academic Editor: S.ย Zhang

Copyright ยฉ 2012 N. Magesh et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We introduce and study a subclass ฮฃ๐‘ƒ(๐›พ,๐‘˜,๐œ†,๐‘) of meromorphic univalent functions defined by certain linear operator involving the generalized hypergeometric function. We obtain coefficient estimates, extreme points, growth and distortion inequalities, radii of meromorphic starlikeness, and convexity for the class ฮฃ๐‘ƒ(๐›พ,๐‘˜,๐œ†,๐‘) by fixing the second coefficient. Further, it is shown that the class ฮฃ๐‘ƒ(๐›พ,๐‘˜,๐œ†) is closed under convex linear combination.

1. Introduction

Let ฮฃ denote the class of functions of the form ๐‘“(๐‘ง)=๐‘งโˆ’1+โˆž๎“๐‘›=1๐‘Ž๐‘›๐‘ง๐‘›,(1.1) which are analytic in the punctured unit disk๐•Œโˆ—โˆถ={๐‘งโˆถ๐‘งโˆˆโ„‚,0<|๐‘ง|<1}=โˆถ๐•Œโงต{0}.(1.2)

Let ฮฃ๐’ฎ,โ€‰โ€‰ฮฃโˆ—(๐›พ) and ฮฃ๐พ(๐›พ),โ€‰โ€‰(0โ‰ค๐›พ<1) denote the subclasses of ฮฃ that are meromorphically univalent, meromorphically starlike functions of order ๐›พ, and meromorphically convex functions of order ๐›พ, respectively. Analytically, ๐‘“โˆˆฮฃโˆ—(๐›พ) if and only if, ๐‘“ is of the form (1.1) and satisfies๎‚ปโˆ’โ„œ๐‘ง๐‘“๎…ž(๐‘ง)๎‚ผ๐‘“(๐‘ง)>๐›พ,๐‘งโˆˆ๐•Œ,(1.3) similarly, ๐‘“โˆˆฮฃ๐พ(๐›พ), if and only if, ๐‘“ is of the form (1.1) and satisfies ๎‚ปโˆ’โ„œ1+๐‘ง๐‘“๎…ž๎…ž(๐‘ง)๐‘“๎…ž๎‚ผ(๐‘ง)>๐›พ,๐‘งโˆˆ๐•Œ,(1.4) and similar other classes of meromorphically univalent functions have been extensively studied by Aouf and Darwish [1], Aouf and Joshi [2], Mogra et al. [3], Uralegaddi and Ganigi [4], Uralegaddi and Somanatha[5], and Owa and Pascu [6].

Let ๐‘“, ๐‘”โˆˆฮฃ, where ๐‘“(๐‘ง) is given by (1.1) and ๐‘”(๐‘ง) is defined by๐‘”(๐‘ง)=๐‘งโˆ’1+โˆž๎“๐‘›=1๐‘๐‘›๐‘ง๐‘›.(1.5)

Then the Hadamard product (or convolution) ๐‘“โˆ—๐‘” of the functions ๐‘“(๐‘ง) and ๐‘”(๐‘ง) is defined by(๐‘“โˆ—๐‘”)(๐‘ง)โˆถ=๐‘งโˆ’1+โˆž๎“๐‘›=1๐‘Ž๐‘›๐‘๐‘›๐‘ง๐‘›=โˆถ(๐‘”โˆ—๐‘“)(๐‘ง).(1.6) Let ฮฃ๐‘ƒ be the class of functions of the form ๐‘“(๐‘ง)=๐‘งโˆ’1+โˆž๎“๐‘›=1๐‘Ž๐‘›๐‘ง๐‘›,๐‘Ž๐‘›โ‰ฅ0,(1.7) that are analytic and univalent in ๐•Œโˆ—.

For complex parameters ๐›ผ1,โ€ฆ,๐›ผ๐‘™ and ๐›ฝ1,โ€ฆ,๐›ฝ๐‘š(๐›ฝ๐‘—โ‰ 0,โˆ’1,โ€ฆ;๐‘—=1,2,โ€ฆ,๐‘š) the generalized hypergeometric function ๐‘™๐น๐‘š(๐‘ง) is defined by ๐‘™๐น๐‘š(๐‘ง)โ‰ก๐‘™๐น๐‘š๎€ท๐›ผ1,โ€ฆ๐›ผ๐‘™;๐›ฝ1,โ€ฆ,๐›ฝ๐‘š๎€ธ;๐‘งโˆถ=โˆž๎“๐‘›=0๎€ท๐›ผ1๎€ธ๐‘›โ€ฆ๎€ท๐›ผ๐‘™๎€ธ๐‘›๎€ท๐›ฝ1๎€ธ๐‘›โ€ฆ๎€ท๐›ฝ๐‘š๎€ธ๐‘›๐‘ง๐‘›,๎€ท๐‘›!๐‘™โ‰ค๐‘š+1;๐‘™,๐‘šโˆˆโ„•0๎€ธ,โˆถ=โ„•โˆช{0};๐‘งโˆˆ๐‘ˆ(1.8) where โ„• denotes the set of all positive integers and (๐œƒ)๐‘› is the Pochhammer symbol defined by(๐œƒ)๐‘›=ฮ“(๐œƒ+๐‘›)=๎‚ปฮ“(๐œƒ)1,๐‘›=0;๐œƒโˆˆโ„‚โงต{0},๐œƒ(๐œƒ+1)(๐œƒ+2)โ‹…(๐œƒ+๐‘›โˆ’1),๐‘›โˆˆโ„•;๐œƒโˆˆโ„‚.(1.9)

Corresponding to a function ๐‘™๐น๐‘š(๐›ผ1,โ€ฆ๐›ผ๐‘™;๐›ฝ1,โ€ฆ,๐›ฝ๐‘š;๐‘ง) defined byโ„ฑ๎€ท๐›ผ1,โ€ฆ๐›ผ๐‘™;๐›ฝ1,โ€ฆ,๐›ฝ๐‘š๎€ธ;๐‘งโˆถ=๐‘ง๐‘™โˆ’1๐น๐‘š๎€ท๐›ผ1,โ€ฆ๐›ผ๐‘™;๐›ฝ1,โ€ฆ,๐›ฝ๐‘š๎€ธ.;๐‘ง(1.10)

Liu and Srivastava [7] considered a linear operator โ„‹(๐›ผ1,โ€ฆ๐›ผ๐‘™;๐›ฝ1,โ€ฆ,๐›ฝ๐‘š)โˆถฮฃโ†’ฮฃ defined by the following Hadamard product (or convolution):โ„‹๎€ท๐›ผ1,โ€ฆ๐›ผ๐‘™;๐›ฝ1,โ€ฆ,๐›ฝ๐‘š๎€ธ๐‘“๎€ท๐›ผ(๐‘ง)=โ„ฑ1,โ€ฆ๐›ผ๐‘™;๐›ฝ1,โ€ฆ,๐›ฝ๐‘š๎€ธ;๐‘งโˆ—๐‘“(๐‘ง)=๐‘งโˆ’1+โˆž๎“๐‘›=1|||||๎€ท๐›ผ1๎€ธ๐‘›+1โ€ฆ๎€ท๐›ผ๐‘™๎€ธ๐‘›+1๎€ท๐›ฝ1๎€ธ๐‘›+1โ€ฆ๎€ท๐›ฝ๐‘š๎€ธ๐‘›+1|||||๐‘Ž๐‘›๐‘ง๐‘›,(๐‘›+1)!(1.11) where, ๐›ผ๐‘–>0, (๐‘–=1,2,โ€ฆ๐‘™), ๐›ฝ๐‘—>0, (๐‘—=1,2,โ€ฆ๐‘š), ๐‘™โ‰ค๐‘š+1;โ€‰โ€‰๐‘™, ๐‘šโˆˆโ„•0=โ„•โˆช{0}. For notational simplicity, we use a shorter notations โ„‹๐‘™๐‘š[๐›ผ1] for โ„‹(๐›ผ1,โ€ฆ๐›ผ๐‘™;๐›ฝ1,โ€ฆ,๐›ฝ๐‘š) and ฮ“๐‘›๎€ท๐›ผ1๎€ธ=๎€ท๐›ผ1๎€ธ๐‘›+1โ€ฆ๎€ท๐›ผ๐‘™๎€ธ๐‘›+1๎€ท๐›ฝ1๎€ธ๐‘›+1โ€ฆ๎€ท๐›ฝ๐‘š๎€ธ๐‘›+11(๐‘›+1)!,(1.12) unless otherwise stated in the sequel. We note that the linear operator ๐ป๐‘™๐‘š[๐›ผ1] was earlier defined for multivalent functions by Dziok and Srivastava [8] and was investigated by Liu and Srivastava [7].

Making use of the operator โ„‹๐‘™๐‘š[๐›ผ1], now we consider a subclass of functions in ฮฃ๐‘ƒ as follows.

Definition 1.1. For 0โ‰ค๐›พ<1 and ๐‘˜โ‰ฅ0 and 0โ‰ค๐œ†<1/2, let ฮฃ(๐›พ,๐‘˜,๐œ†) denote a subclass of ฮฃ consisting functions of the form (1.1) satisfying the condition that ๎ƒฏ๐‘ง๎€ทโ„‹๎€บ๐›ผโˆ’โ„œ1๎€ป๎€ธ๐‘“(๐‘ง)โ€ฒ+๐œ†๐‘ง2๎€ทโ„‹๎€บ๐›ผ1๎€ป๎€ธ๐‘“(๐‘ง)๎…ž๎…ž๎€บ๐›ผ(1โˆ’๐œ†)โ„‹1๎€ป๐‘“๎€ทโ„‹๎€บ๐›ผ(๐‘ง)+๐œ†๐‘ง1๎€ป๐‘“๎€ธโ€ฒ๎ƒฐ|||||๐‘ง๎€ทโ„‹๎€บ๐›ผ(๐‘ง)+๐›พ>๐‘˜1๎€ป๎€ธ๐‘“(๐‘ง)โ€ฒ+๐œ†๐‘ง2๎€ทโ„‹๎€บ๐›ผ1๎€ป๎€ธ๐‘“(๐‘ง)๎…ž๎…ž๎€บ๐›ผ(1โˆ’๐œ†)โ„‹1๎€ป๎€ทโ„‹๎€บ๐›ผ๐‘“(๐‘ง)+๐œ†๐‘ง1๎€ป๎€ธโ€ฒ|||||๐‘“(๐‘ง)+1,๐‘งโˆˆ๐•Œโˆ—,(1.13) where โ„‹[๐›ผ1]๐‘“(๐‘ง) is given by (1.11). Furthermore, we say that a function ๐‘“โˆˆฮฃ๐‘ƒ(๐›พ,๐‘˜,๐œ†), whenever ๐‘“(๐‘ง) is of the form (1.7).
We observe that, by specializing the parameters ๐‘™,โ€‰โ€‰๐‘š,โ€‰โ€‰๐›ผ1,โ€ฆ,๐›ผ๐‘™,โ€‰โ€‰๐›ฝ1,โ€ฆ,๐›ฝ๐‘š,โ€‰โ€‰๐‘˜,โ€‰โ€‰๐›พ and ๐œ† the class leads to various subclasses. As for illustrations, we present some examples for the cases.

Example 1.2. If ๐‘™=2 and ๐‘š=1 with ๐›ผ1=1, ๐›ผ2=1, ๐›ฝ1=1 andโ€‰โ€‰๐‘“(๐‘ง)โ€‰โ€‰of the form (1.7), then we obtain the new subclass โ„ณ๐‘ƒ(๐›พ,๐‘˜,๐œ†) โ€‰โ€‰defined by ๎‚ปโˆ’โ„œ๐‘ง๐‘“๎…ž(๐‘ง)+๐œ†๐‘ง2๐‘“๎…ž๎…ž(๐‘ง)(1โˆ’๐œ†)๐‘“(๐‘ง)+๐œ†๐‘ง๐‘“๎…ž๎‚ผ||||(๐‘ง)+๐›พ>๐‘˜๐‘ง๐‘“๎…ž(๐‘ง)+๐œ†๐‘ง2๐‘“๎…ž๎…ž(๐‘ง)(1โˆ’๐œ†)๐‘“(๐‘ง)+๐œ†๐‘ง๐‘“๎…ž||||(๐‘ง)+1,๐‘งโˆˆ๐•Œโˆ—.(1.14)

Example 1.3. For ๐‘™=2, ๐‘š=1, ๐›ผ1=๐›ฟ+1, ๐›ฝ1=๐›ผ2=1, and๐‘“(๐‘ง) of the form (1.7), then we get the new subclass ๐’Ÿ๐›ฟ๐‘ƒ(๐›พ,๐‘˜,๐œ†) defined by โŽงโŽชโŽจโŽชโŽฉ๐‘ง๎€ท๐’Ÿโˆ’โ„œ๐›ฟ๎€ธ๐‘“(๐‘ง)๎…ž+๐œ†๐‘ง2๎€ท๐’Ÿ๐›ฟ๎€ธ๐‘“(๐‘ง)๎…ž๎…ž(1โˆ’๐œ†)๐’Ÿ๐›ฟ๎€ท๐’Ÿ๐‘“(๐‘ง)+๐œ†๐‘ง๐›ฟ๎€ธโ€ฒโŽซโŽชโŽฌโŽชโŽญ|||||๐‘ง๎€ท๐’Ÿ๐‘“(๐‘ง)+๐›พ>๐‘˜๐›ฟ๎€ธ๐‘“(๐‘ง)๎…ž+๐œ†๐‘ง2๎€ท๐’Ÿ๐›ฟ๎€ธ๐‘“(๐‘ง)๎…ž๎…ž(1โˆ’๐œ†)๐’Ÿ๐›ฟ๎€ท๐’Ÿ๐‘“(๐‘ง)+๐œ†๐‘ง๐›ฟ๎€ธโ€ฒ|||||๐‘“(๐‘ง)+1,๐‘งโˆˆ๐•Œโˆ—,(1.15) where ๐’Ÿ๐›ฟ๐‘“(๐‘ง)=1/(๐‘ง(1โˆ’๐‘ง)๐›ฟ+1)โˆ—๐‘“(๐‘ง)(๐›ฟ>โˆ’1), is the differential operator which was introduced by Ganigi and Uralegaddi [9]. Also we note that the class ๐’Ÿ๐›ฟ๐‘ƒ(๐›พ,๐‘˜,๐œ†) was introduced by Atshan and Kulkarni [10].

Example 1.4. For ๐‘™=2, ๐‘š=1,โ€‰โ€‰๐›ผ2=1, and ๐‘“(๐‘ง) of the form (1.7), then we obtain the new subclass โ„’๐‘ƒ(๐›พ,๐‘˜,๐œ†) defined by ๎ƒฏ๐‘ง๎€ทโ„’๎€บ๐›ผโˆ’โ„œ1,๐›ฝ1๎€ป๎€ธ๐‘“(๐‘ง)โ€ฒ+๐œ†๐‘ง2๎€ทโ„’๎€บ๐›ผ1,๐›ฝ1๎€ป๎€ธ๐‘“(๐‘ง)๎…ž๎…ž๎€บ๐›ผ(1โˆ’๐œ†)โ„’1,๐›ฝ1๎€ป๐‘“๎€ทโ„’๎€บ๐›ผ(๐‘ง)+๐œ†๐‘ง1,๐›ฝ1๎€ป๐‘“๎€ธโ€ฒ๎ƒฐ|||||๐‘ง๎€ทโ„’๎€บ๐›ผ(๐‘ง)+๐›พ>๐‘˜1,๐›ฝ1๎€ป๎€ธ๐‘“(๐‘ง)โ€ฒ+๐œ†๐‘ง2๎€ทโ„’๎€บ๐›ผ1,๐›ฝ1๎€ป๎€ธ๐‘“(๐‘ง)๎…ž๎…ž๎€บ๐›ผ(1โˆ’๐œ†)โ„’1,๐›ฝ1๎€ป๎€ทโ„’๎€บ๐›ผ๐‘“(๐‘ง)+๐œ†๐‘ง1,๐›ฝ1๎€ป๎€ธโ€ฒ|||||๐‘“(๐‘ง)+1,๐‘งโˆˆ๐•Œโˆ—,(1.16) where the operator โ„’[๐›ผ1;๐›ฝ1]๐‘“(๐‘ง) was introduced and studied by Liu and Srivastava [11] (see also [12, 13]).
For the class ฮฃ๐‘ƒ(๐›พ,๐‘˜,๐œ†), the following characterization was given by Magesh et al. [14].

Theorem 1.5. Let ๐‘“โˆˆฮฃ๐‘ƒ be given by (1.7). Then ๐‘“โˆˆฮฃ๐‘ƒ(๐›พ,๐‘˜,๐œ†), if and only if, โˆž๎“๐‘›=1([]ฮ“1+(๐‘›โˆ’1)๐œ†)๐‘›(๐‘˜+1)+(๐‘˜+๐›พ)๐‘›๎€ท๐›ผ1๎€ธ๐‘Ž๐‘›โ‰ค(1โˆ’๐›พ)(1โˆ’2๐œ†),(1.17) where ฮ“๐‘›(๐›ผ1) is given by (1.12).

For a function defined by (1.7) and in the class ฮฃ๐‘ƒ(๐›พ,๐‘˜,๐œ†), Theorem 1.5, immediately yieldsฮ“1๎€ท๐›ผ1๎€ธ๐‘Ž1โ‰ค(1โˆ’๐›พ)(1โˆ’2๐œ†)(2๐‘˜+๐›พ+1).(1.18) Hence we may takeฮ“1๎€ท๐›ผ1๎€ธ๐‘Ž1=(1โˆ’๐›พ)(1โˆ’2๐œ†)๐‘(2๐‘˜+๐›พ+1),0<๐‘<1.(1.19)

Motivated by Aouf and Darwish [1], Aouf and Joshi [2], Ghanim and Darus [12], and Uralegaddi [15], we now introduce the following class of functions and use the similar techniques to prove our results.

Let ฮฃ๐‘ƒ(๐›พ,๐‘˜,๐œ†,๐‘) be the subclass of ฮฃ๐‘ƒ(๐›พ,๐‘˜,๐œ†) consisting functions in of the form1๐‘“(๐‘ง)=๐‘ง+(1โˆ’๐›พ)(1โˆ’2๐œ†)๐‘(2๐‘˜+๐›พ+1)๐‘ง+โˆž๎“๐‘›=2ฮ“๐‘›๎€ท๐›ผ1๎€ธ๐‘Ž๐‘›๐‘ง๐‘›,0<๐‘<1.(1.20)

In this paper, coefficient estimates, extreme points, growth and distortion bounds, radii of meromorphically starlikeness and convexity are discussed for the class ฮฃ๐‘ƒ(๐›พ,๐‘˜,๐œ†) by fixing the second coefficient. Further, it is shown that the class ฮฃ๐‘ƒ(๐›พ,๐‘˜,๐œ†) is closed under convex linear combination.

2. Main Results

In our first theorem, we now find out the coefficient inequality for the class ฮฃ๐‘ƒ(๐›พ,๐‘˜,๐œ†,๐‘).

Theorem 2.1. Let the function ๐‘“(๐‘ง) defined by (1.20). Then ๐‘“โˆˆฮฃ๐‘ƒ(๐›พ,๐‘˜,๐œ†,๐‘), if and only if, โˆž๎“๐‘›=2([]ฮ“1+(๐‘›โˆ’1)๐œ†)๐‘›(๐‘˜+1)+(๐‘˜+๐›พ)๐‘›๎€ท๐›ผ1๎€ธ๐‘Ž๐‘›โ‰ค(1โˆ’๐›พ)(1โˆ’2๐œ†)(1โˆ’๐‘).(2.1) The result is sharp.

Proof. By putting ฮ“1๎€ท๐›ผ1๎€ธ๐‘Ž1=(1โˆ’๐›พ)(1โˆ’2๐œ†)๐‘(2๐‘˜+๐›พ+1),0<๐‘<1,(2.2) in (1.17), the result is easily derived. The result is sharp for the function 1๐‘“(๐‘ง)=๐‘ง+(1โˆ’๐›พ)(1โˆ’2๐œ†)๐‘(2๐‘˜+๐›พ+1)๐‘ง+(1โˆ’๐›พ)(1โˆ’2๐œ†)(1โˆ’๐‘)[]ฮ“(1+(๐‘›โˆ’1)๐œ†)๐‘›(๐‘˜+1)+(๐‘˜+๐›พ)๐‘›๎€ท๐›ผ1๎€ธ๐‘ง๐‘›,๐‘›โ‰ฅ2.(2.3)

Corollary 2.2. If the function ๐‘“ defined by (1.20) is in the class ฮฃ๐‘ƒ(๐›พ,๐‘˜,๐œ†,๐‘), then ๐‘Ž๐‘›โ‰ค(1โˆ’๐›พ)(1โˆ’2๐œ†)(1โˆ’๐‘)[]ฮ“(1+(๐‘›โˆ’1)๐œ†)๐‘›(๐‘˜+1)+(๐‘˜+๐›พ)๐‘›๎€ท๐›ผ1๎€ธ,๐‘›โ‰ฅ2.(2.4) The result is sharp for the function ๐‘“(๐‘ง) given by (2.3).

Next we prove the following growth and distortion properties for the class ฮฃ๐‘ƒ(๐›พ,๐‘˜,๐œ†,๐‘).

Theorem 2.3. If the function ๐‘“(๐‘ง) defined by (1.20) is in the class ฮฃ๐‘ƒ(๐›พ,๐‘˜,๐œ†,๐‘) for 0<|๐‘ง|=๐‘Ÿ<1, then one has 1๐‘Ÿโˆ’(1โˆ’๐›พ)(1โˆ’2๐œ†)๐‘(2๐‘˜+๐›พ+1)๐‘Ÿโˆ’(1โˆ’๐›พ)(1โˆ’2๐œ†)(1โˆ’๐‘)๐‘Ÿ(1+๐œ†)(3๐‘˜+๐›พ+2)2โ‰ค||||โ‰ค1๐‘“(๐‘ง)๐‘Ÿ+(1โˆ’๐›พ)(1โˆ’2๐œ†)๐‘(2๐‘˜+๐›พ+1)๐‘Ÿ+(1โˆ’๐›พ)(1โˆ’2๐œ†)(1โˆ’๐‘)๐‘Ÿ(1+๐œ†)(3๐‘˜+๐›พ+2)2.(2.5) The result is sharp for the function ๐‘“(๐‘ง) given by 1๐‘“(๐‘ง)=๐‘ง+(1โˆ’๐›พ)(1โˆ’2๐œ†)๐‘(2๐‘˜+๐›พ+1)๐‘ง+(1โˆ’๐›พ)(1โˆ’2๐œ†)(1โˆ’๐‘)๐‘ง(1+๐œ†)(3๐‘˜+๐›พ+2)2.(2.6)

Proof. Since ฮฃ๐‘ƒ(๐›พ,๐‘˜,๐œ†,๐‘), Theorem 2.1 yields ฮ“๐‘›๎€ท๐›ผ1๎€ธ๐‘Ž๐‘›โ‰ค(1โˆ’๐›พ)(1โˆ’2๐œ†)(1โˆ’๐‘)[](1+(๐‘›โˆ’1)๐œ†)๐‘›(๐‘˜+1)+(๐‘˜+๐›พ),๐‘›โ‰ฅ2.(2.7) Thus, for 0<|๐‘ง|=๐‘Ÿ<1, ||||โ‰ค1๐‘“(๐‘ง)+|๐‘ง|(1โˆ’๐›พ)(1โˆ’2๐œ†)๐‘|(2๐‘˜+๐›พ+1)๐‘ง|+โˆž๎“๐‘›=2ฮ“๐‘›๎€ท๐›ผ1๎€ธ๐‘Ž๐‘›|๐‘ง|๐‘›โ‰ค1๐‘Ÿ+(1โˆ’๐›พ)(1โˆ’2๐œ†)๐‘(2๐‘˜+๐›พ+1)๐‘Ÿ+๐‘Ÿ2โˆž๎“๐‘›=2ฮ“๐‘›๎€ท๐›ผ1๎€ธ๐‘Ž๐‘›โ‰ค1๐‘Ÿ+(1โˆ’๐›พ)(1โˆ’2๐œ†)๐‘(2๐‘˜+๐›พ+1)๐‘Ÿ+(1โˆ’๐›พ)(1โˆ’2๐œ†)(1โˆ’๐‘)๐‘Ÿ(1+๐œ†)(3๐‘˜+๐›พ+2)2,||||โ‰ฅ1๐‘“(๐‘ง)โˆ’|๐‘ง|(1โˆ’๐›พ)(1โˆ’2๐œ†)๐‘(2๐‘˜+๐›พ+1)|๐‘ง|โˆ’โˆž๎“๐‘›=2ฮ“๐‘›๎€ท๐›ผ1๎€ธ๐‘Ž๐‘›|๐‘ง|๐‘›โ‰ฅ1๐‘Ÿโˆ’(1โˆ’๐›พ)(1โˆ’2๐œ†)๐‘(2๐‘˜+๐›พ+1)๐‘Ÿโˆ’๐‘Ÿ2โˆž๎“๐‘›=2ฮ“๐‘›๎€ท๐›ผ1๎€ธ๐‘Ž๐‘›โ‰ฅ1๐‘Ÿโˆ’(1โˆ’๐›พ)(1โˆ’2๐œ†)๐‘(2๐‘˜+๐›พ+1)๐‘Ÿโˆ’(1โˆ’๐›พ)(1โˆ’2๐œ†)(1โˆ’๐‘)๐‘Ÿ(1+๐œ†)(3๐‘˜+๐›พ+2)2.(2.8) Thus the proof of the theorem is complete.

Theorem 2.4. If the function ๐‘“(๐‘ง) defined by (1.20) is in the class ฮฃ๐‘ƒ(๐›พ,๐‘˜,๐œ†,๐‘) for 0<|๐‘ง|=๐‘Ÿ<1, then one has 1๐‘Ÿ2โˆ’(1โˆ’๐›พ)(1โˆ’2๐œ†)๐‘โˆ’(2๐‘˜+๐›พ+1)(1โˆ’๐›พ)(1โˆ’2๐œ†)(1โˆ’๐‘)๐‘Ÿโ‰ค||๐‘“(1+๐œ†)(3๐‘˜+๐›พ+2)๎…ž||โ‰ค1(๐‘ง)๐‘Ÿ2+(1โˆ’๐›พ)(1โˆ’2๐œ†)๐‘+(2๐‘˜+๐›พ+1)(1โˆ’๐›พ)(1โˆ’2๐œ†)(1โˆ’๐‘)(1+๐œ†)(3๐‘˜+๐›พ+2)๐‘Ÿ.(2.9) The result is sharp for the function ๐‘“(๐‘ง) given by 1๐‘“(๐‘ง)=๐‘ง+(1โˆ’๐›พ)(1โˆ’2๐œ†)๐‘(2๐‘˜+๐›พ+1)๐‘ง+(1โˆ’๐›พ)(1โˆ’2๐œ†)(1โˆ’๐‘)๐‘ง(1+๐œ†)(3๐‘˜+๐›พ+2)2.(2.10)

Proof. In view of Theorem 2.1, it follows that ๐‘›ฮ“๐‘›๎€ท๐›ผ1๎€ธ๐‘Ž๐‘›โ‰ค๐‘›(1โˆ’๐›พ)(1โˆ’2๐œ†)(1โˆ’๐‘)[](1+(๐‘›โˆ’1)๐œ†)๐‘›(๐‘˜+1)+(๐‘˜+๐›พ),๐‘›โ‰ฅ2.(2.11) Thus, for 0<|๐‘ง|=๐‘Ÿ<1 and making use of (2.11), we obtain ||๐‘“๎…ž(||โ‰ค|||๐‘ง)โˆ’1๐‘ง2|||+(1โˆ’๐›พ)(1โˆ’2๐œ†)๐‘+(2๐‘˜+๐›พ+1)โˆž๎“๐‘›=2๐‘›ฮ“๐‘›๎€ท๐›ผ1๎€ธ๐‘Ž๐‘›|๐‘ง|๐‘›โˆ’1โ‰ค1๐‘Ÿ2+(1โˆ’๐›พ)(1โˆ’2๐œ†)๐‘(2๐‘˜+๐›พ+1)+๐‘Ÿโˆž๎“๐‘›=2๐‘›ฮ“๐‘›๎€ท๐›ผ1๎€ธ๐‘Ž๐‘›โ‰ค1๐‘Ÿ2+(1โˆ’๐›พ)(1โˆ’2๐œ†)๐‘+(2๐‘˜+๐›พ+1)(1โˆ’๐›พ)(1โˆ’2๐œ†)(1โˆ’๐‘)||๐‘“(1+๐œ†)(3๐‘˜+๐›พ+2)๐‘Ÿ,๎…ž||โ‰ฅ|||(๐‘ง)โˆ’1๐‘ง2|||โˆ’(1โˆ’๐›พ)(1โˆ’2๐œ†)๐‘โˆ’(2๐‘˜+๐›พ+1)โˆž๎“๐‘›=2๐‘›ฮ“๐‘›๎€ท๐›ผ1๎€ธ๐‘Ž๐‘›|๐‘ง|๐‘›โˆ’1โ‰ฅ1๐‘Ÿ2โˆ’(1โˆ’๐›พ)(1โˆ’2๐œ†)๐‘(2๐‘˜+๐›พ+1)โˆ’๐‘Ÿโˆž๎“๐‘›=2๐‘›ฮ“๐‘›๎€ท๐›ผ1๎€ธ๐‘Ž๐‘›โ‰ฅ1๐‘Ÿ2โˆ’(1โˆ’๐›พ)(1โˆ’2๐œ†)๐‘โˆ’(2๐‘˜+๐›พ+1)(1โˆ’๐›พ)(1โˆ’2๐œ†)(1โˆ’๐‘)(1+๐œ†)(3๐‘˜+๐›พ+2)๐‘Ÿ.(2.12)

Hence the result follows.

Next, we will show that the class ฮฃ๐‘ƒ(๐›พ,๐‘˜,๐œ†,๐‘) is closed under convex linear combination.

Theorem 2.5. If ๐‘“1(1๐‘ง)=๐‘ง+(1โˆ’๐›พ)(1โˆ’2๐œ†)๐‘(2๐‘˜+๐›พ+1)๐‘ง,(2.13) and for ๐‘›โ‰ฅ2๐‘“๐‘›(1๐‘ง)=๐‘ง+(1โˆ’๐›พ)(1โˆ’2๐œ†)๐‘(2๐‘˜+๐›พ+1)๐‘ง+โˆž๎“๐‘›=2(1โˆ’๐›พ)(1โˆ’2๐œ†)(1โˆ’๐‘)[]ฮ“(1+(๐‘›โˆ’1)๐œ†)๐‘›(๐‘˜+1)+(๐‘˜+๐›พ)๐‘›๎€ท๐›ผ1๎€ธ๐‘ง๐‘›.(2.14) Then ๐‘“โˆˆฮฃ๐‘ƒ(๐›พ,๐‘˜,๐œ†,๐‘) if and only if it can expressed in the form ๐‘“(๐‘ง)=โˆž๎“๐‘›=1๐œ‡๐‘›๐‘“๐‘›(๐‘ง),(2.15) where ๐œ‡๐‘›โ‰ฅ0 and โˆ‘โˆž๐‘›=1๐œ‡๐‘›โ‰ค1.

Proof. From (2.13), (2.14), and (2.15), we have 1๐‘“(๐‘ง)=๐‘ง+(1โˆ’๐›พ)(1โˆ’2๐œ†)๐‘(2๐‘˜+๐›พ+1)๐‘ง+โˆž๎“๐‘›=2(1โˆ’๐›พ)(1โˆ’2๐œ†)(1โˆ’๐‘)๐œ‡๐‘›[]ฮ“(1+(๐‘›โˆ’1)๐œ†)๐‘›(๐‘˜+1)+(๐‘˜+๐›พ)๐‘›๎€ท๐›ผ1๎€ธ๐‘ง๐‘›.(2.16) Since โˆž๎“๐‘›=2(1โˆ’๐›พ)(1โˆ’2๐œ†)(1โˆ’๐‘)๐œ‡๐‘›[]ฮ“(1+(๐‘›โˆ’1)๐œ†)๐‘›(๐‘˜+1)+(๐‘˜+๐›พ)๐‘›๎€ท๐›ผ1๎€ธ[]ฮ“(1+(๐‘›โˆ’1)๐œ†)๐‘›(๐‘˜+1)+(๐‘˜+๐›พ)๐‘›๎€ท๐›ผ1๎€ธ=(1โˆ’๐›พ)(1โˆ’2๐œ†)(1โˆ’๐‘)โˆž๎“๐‘›=2๐œ‡๐‘›=1โˆ’๐œ‡1โ‰ค1,(2.17) it follows from Theorem 1.5 that the function ๐‘“โˆˆฮฃ๐‘ƒ(๐›พ,๐‘˜,๐œ†,๐‘). Conversely, suppose that ๐‘“โˆˆฮฃ๐‘ƒ(๐›พ,๐‘˜,๐œ†,๐‘). Since ๐‘Ž๐‘›โ‰ค(1โˆ’๐›พ)(1โˆ’2๐œ†)(1โˆ’๐‘)[]ฮ“(1+(๐‘›โˆ’1)๐œ†)๐‘›(๐‘˜+1)+(๐‘˜+๐›พ)๐‘›๎€ท๐›ผ1๎€ธ,๐‘›โ‰ฅ2.(2.18) Setting ๐œ‡๐‘›=[๐‘›]ฮ“(1+(๐‘›โˆ’1)๐œ†)(๐‘˜+1)+(๐‘˜+๐›พ)๐‘›๎€ท๐›ผ1๎€ธ๐‘Ž(1โˆ’๐›พ)(1โˆ’2๐œ†)(1โˆ’๐‘)๐‘›,๐œ‡1=1โˆ’โˆž๎“๐‘›=2๐œ‡๐‘›.(2.19) it follows that ๐‘“(๐‘ง)=โˆž๎“๐‘›=1๐œ‡๐‘›๐‘“๐‘›(๐‘ง).(2.20) This completes the proof of the theorem.

Theorem 2.6. The class ฮฃ๐‘ƒ(๐›พ,๐‘˜,๐œ†,๐‘) is closed under linear combination.

Proof. Suppose that the function ๐‘“ be given by (1.20), and let the function ๐‘” be given by 1๐‘”(๐‘ง)=๐‘ง+(1โˆ’๐›พ)(1โˆ’2๐œ†)๐‘(2๐‘˜+๐›พ+1)๐‘ง+โˆž๎“๐‘›=2||๐‘๐‘›||๐‘ง๐‘›,๐‘›โ‰ฅ2.(2.21) Assuming that ๐‘“ and ๐‘” are in the class ฮฃ๐‘ƒ(๐›พ,๐‘˜,๐œ†,๐‘), it is enough to prove that the function โ„Ž defined by โ„Ž(๐‘ง)=๐œ‡๐‘“(๐‘ง)+(1โˆ’๐œ‡)๐‘”(๐‘ง),0โ‰ค๐œ‡โ‰ค1,(2.22) is also in the class ฮฃ๐‘ƒ(๐›พ,๐‘˜,๐œ†,๐‘). Since 1โ„Ž(๐‘ง)=๐‘ง+(1โˆ’๐›พ)(1โˆ’2๐œ†)๐‘(2๐‘˜+๐›พ+1)๐‘ง+โˆž๎“๐‘›=2||๐‘Ž๐‘›๐œ‡+(1โˆ’๐œ‡)๐‘๐‘›||๐‘ง๐‘›,(2.23) we observe that โˆž๎“๐‘›=2([]ฮ“1+(๐‘›โˆ’1)๐œ†)๐‘›(๐‘˜+1)+(๐‘˜+๐›พ)๐‘›๎€ท๐›ผ1๎€ธ||๐‘Ž๐‘›๐œ‡+(1โˆ’๐œ‡)๐‘๐‘›||โ‰ค(1โˆ’๐›พ)(1โˆ’2๐œ†)(1โˆ’๐‘),(2.24) with the aid of Theorem 2.1. Thus, โ„Žโˆˆฮฃ๐‘ƒ(๐›พ,๐‘˜,๐œ†,๐‘).

Next we determine the radii of meromorphically starlikeness and convexity of order ๐›ฟ for functions in the class ฮฃ๐‘ƒ(๐›พ,๐‘˜,๐œ†,๐‘).

Theorem 2.7. Let the function ๐‘“(๐‘ง) defined by (1.20) be in the class ฮฃ๐‘ƒ(๐›พ,๐‘˜,๐œ†,๐‘), then (i)๐‘“ is meromorphically starlike of order ๐›ฟ(0โ‰ค๐›ฟ<1) in the disk |๐‘ง|<๐‘Ÿ1(๐›พ,๐‘˜,๐œ†,๐‘,๐›ฟ) where ๐‘Ÿ1(๐›พ,๐‘˜,๐œ†,๐‘,๐›ฟ), is the largest value for which (3โˆ’๐›ฟ)(1โˆ’๐›พ)(1โˆ’2๐œ†)๐‘๐‘Ÿ(2๐‘˜+๐›พ+1)2+(๐‘›+2โˆ’๐›ฟ)(1โˆ’๐›พ)(1โˆ’2๐œ†)(1โˆ’๐‘)[]๐‘Ÿ(1+(๐‘›โˆ’1)๐œ†)๐‘›(๐‘˜+1)+(๐‘˜+๐›พ)๐‘›+1โ‰ค(1โˆ’๐›ฟ),๐‘›โ‰ฅ2,(2.25)(ii)๐‘“ is meromorphically convex of order ๐›ฟ(0โ‰ค๐›ฟ<1) in the disk |๐‘ง|<๐‘Ÿ2(๐›พ,๐‘˜,๐œ†,๐‘,๐›ฟ) where ๐‘Ÿ2(๐›พ,๐‘˜,๐œ†,๐‘,๐›ฟ), is the largest value for which(3โˆ’๐›ฟ)(1โˆ’๐›พ)(1โˆ’2๐œ†)๐‘๐‘Ÿ(2๐‘˜+๐›พ+1)2+๐‘›(๐‘›+2โˆ’๐›ฟ)(1โˆ’๐›พ)(1โˆ’2๐œ†)(1โˆ’๐‘)[]๐‘Ÿ(1+(๐‘›โˆ’1)๐œ†)๐‘›(๐‘˜+1)+(๐‘˜+๐›พ)๐‘›+1โ‰ค(1โˆ’๐›ฟ),๐‘›โ‰ฅ2.(2.26) Each of these results is sharp for function ๐‘“(๐‘ง) given by (2.3).

Proof. It is enough to highlight that ||||๐‘ง๐‘“๎…ž(๐‘ง)||||๐‘“(๐‘ง)+1โ‰ค1โˆ’๐›ฟ,|๐‘ง|<๐‘Ÿ1.(2.27) Thus, we have ||||๐‘ง๐‘“๎…ž(๐‘ง)||||=||||โˆ‘๐‘“(๐‘ง)+1โˆ’1/๐‘ง+((๐‘‰)(๐ธ)๐‘/(๐‘„))๐‘ง+โˆž๐‘›=2๐‘›ฮ“๐‘›๎€ท๐›ผ1๎€ธ๐‘Ž๐‘›๐‘ง๐‘›โˆ‘+1/๐‘ง+((๐‘‰)(๐ธ)๐‘/(๐‘„))๐‘ง+โˆž๐‘›=2ฮ“๐‘›๎€ท๐›ผ1๎€ธ๐‘Ž๐‘›๐‘ง๐‘›โˆ‘1/๐‘ง+((๐‘‰)(๐ธ)๐‘/(๐‘„))๐‘ง+โˆž๐‘›=2ฮ“๐‘›๎€ท๐›ผ1๎€ธ๐‘Ž๐‘›๐‘ง๐‘›||||,(2.28) where (๐‘„) denotes (2๐‘˜+๐›พ+1),๐‘‰ denotes (1โˆ’๐›พ), and (๐ธ) denotes (1โˆ’2๐œ†).
Hence (2.28) holds true if 2(1โˆ’๐›พ)(1โˆ’2๐œ†)๐‘๐‘Ÿ(2๐‘˜+๐›พ+1)2+โˆž๎“๐‘›=2(๐‘›+1)ฮ“๐‘›๎€ท๐›ผ1๎€ธ๐‘Ž๐‘›๐‘Ÿ๐‘›+1๎ƒฌ(โ‰ค(1โˆ’๐›ฟ)1โˆ’1โˆ’๐›พ)(1โˆ’2๐œ†)๐‘๐‘Ÿ(2๐‘˜+๐›พ+1)2โˆ’โˆž๎“๐‘›=2ฮ“๐‘›๎€ท๐›ผ1๎€ธ๐‘Ž๐‘›๐‘Ÿ๐‘›+1๎ƒญ,(2.29) or, (3โˆ’๐›ฟ)(1โˆ’๐›พ)(1โˆ’2๐œ†)๐‘๐‘Ÿ(2๐‘˜+๐›พ+1)2+โˆž๎“๐‘›=2(๐‘›+2โˆ’๐›ฟ)ฮ“๐‘›๎€ท๐›ผ1๎€ธ๐‘Ž๐‘›๐‘Ÿ๐‘›+1โ‰ค(1โˆ’๐›ฟ),(2.30) and it follows that, from (2.1), we may take ๐‘Ž๐‘›โ‰ค(1โˆ’๐›พ)(1โˆ’2๐œ†)(1โˆ’๐‘)[]ฮ“(1+(๐‘›โˆ’1)๐œ†)๐‘›(๐‘˜+1)+(๐‘˜+๐›พ)๐‘›๎€ท๐›ผ1๎€ธ๐œ‡๐‘›,๐‘›โ‰ฅ2,(2.31) where ๐œ‡๐‘›โ‰ฅ0 and โˆ‘โˆž๐‘›=2๐œ‡๐‘›โ‰ค1.
For each fixed ๐‘Ÿ, we choose the positive integer ๐‘›0=๐‘›0(๐‘Ÿ) for which (๐‘›+2โˆ’๐›ฟ)([]ฮ“1+(๐‘›โˆ’1)๐œ†)๐‘›(๐‘˜+1)+(๐‘˜+๐›พ)๐‘›๎€ท๐›ผ1๎€ธ๐‘Ÿ๐‘›+1(2.32) is maximal. Then it follows that โˆž๎“๐‘›=2(๐‘›+2โˆ’๐›ฟ)ฮ“๐‘›๎€ท๐›ผ1๎€ธ๐‘Ž๐‘›๐‘Ÿ๐‘›+1โ‰ค๎€ท๐‘›0๎€ธ+2โˆ’๐›ฟ(1โˆ’๐›พ)(1โˆ’2๐œ†)(1โˆ’๐‘)๎€ท๎€ท๐‘›1+0๎€ธ๐œ†๐‘›โˆ’1๎€ธ๎€บ0๎€ป๐‘Ÿ(๐‘˜+1)+(๐‘˜+๐›พ)๐‘›0+1.(2.33) Then ๐‘“ is starlike of order ๐›ฟ in 0<|๐‘ง|<๐‘Ÿ1(๐›พ,๐‘˜,๐œ†,๐‘,๐›ฟ) provided that (3โˆ’๐›ฟ)(1โˆ’๐›พ)(1โˆ’2๐œ†)๐‘๐‘Ÿ(2๐‘˜+๐›พ+1)2+๎€ท๐‘›0๎€ธ+2โˆ’๐›ฟ(1โˆ’๐›พ)(1โˆ’2๐œ†)(1โˆ’๐‘)๎€ท๎€ท๐‘›1+0๎€ธ๐œ†๐‘›โˆ’1๎€ธ๎€บ0(๎€ป๐‘Ÿ๐‘˜+1)+(๐‘˜+๐›พ)๐‘›0+1โ‰ค(1โˆ’๐›ฟ).(2.34) We find the value ๐‘Ÿ0=๐‘Ÿ0(๐‘˜,๐‘,๐›ฟ,๐‘›) and the corresponding integer ๐‘›0(๐‘Ÿ0) so that (3โˆ’๐›ฟ)(1โˆ’๐›พ)(1โˆ’2๐œ†)๐‘๐‘Ÿ(2๐‘˜+๐›พ+1)2+๎€ท๐‘›0๎€ธ+2โˆ’๐›ฟ(1โˆ’๐›พ)(1โˆ’2๐œ†)(1โˆ’๐‘)๎€ท๎€ท๐‘›1+0๎€ธ๐œ†๐‘›โˆ’1๎€ธ๎€บ0(๎€ป๐‘Ÿ๐‘˜+1)+(๐‘˜+๐›พ)๐‘›0+1=(1โˆ’๐›ฟ).(2.35) It is the value for which the function ๐‘“(๐‘ง) is starlike in 0<|๐‘ง|<๐‘Ÿ0. (ii) In a similar manner, we can prove our result providing the radius of meromorphic convexity of order ๐›ฟ(0โ‰ค๐›ฟ<1) for functions in the class ฮฃ๐‘ƒ(๐›พ,๐‘˜,๐œ†,๐‘), so we skip the details of the proof of (ii).

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