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Abstract and Applied Analysis
Volumeย 2011ย (2011), Article IDย 361647, 15 pages
http://dx.doi.org/10.1155/2011/361647
Research Article

Some Properties of Subclasses of Multivalent Functions

Department of Mathematics, Faculty of Science, Atatรผrk University, 25240 Erzurum, Turkey

Received 8 November 2010; Accepted 8 January 2011

Academic Editor: Ondล™ejย Doลกlรฝ

Copyright ยฉ 2011 Muhammet Kamali and Fatma SaฤŸsรถz. 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

The authors introduce two new subclasses denoted by โ„‘โˆ—(ฮฉ,๐œ†,๐‘,๐›ผ) and โ„‘โˆ—๐œ€(ฮฉ,๐œ†,๐‘,๐›ผ) of the class ๐ด(๐‘,๐‘›) of ๐‘-valent analytic functions. They obtain coefficient inequality for the class โ„‘โˆ—(ฮฉ,๐œ†,๐‘,๐›ผ). They investigate various properties of classes โ„‘โˆ—(ฮฉ,๐œ†,๐‘,๐›ผ) and โ„‘โˆ—๐œ€(ฮฉ,๐œ†,๐‘,๐›ผ). Furthermore, they derive partial sums associated with the class โ„‘โˆ—๐œ€(ฮฉ,๐œ†,๐‘,๐›ผ).

1. Introduction and Definition

Let ๐ด(๐‘,๐‘›) denote the class of functions of the form ๐‘“(๐‘ง)=๐‘ง๐‘+โˆž๎“๐‘˜=๐‘›๐‘Ž๐‘+๐‘˜๐‘ง๐‘+๐‘˜(๐‘›,๐‘โˆˆโ„•={1,2,โ€ฆ}),(1.1) which are analytic and p-valent in the open unit disc ๐‘ˆ={๐‘งโˆถ๐‘งโˆˆโ„‚,|๐‘ง|<1}. We write ๐ด(1,1)=๐ด.

A function ๐‘“โˆˆ๐ด(๐‘,๐‘›) is said to be in the class ๐‘†(๐‘,๐‘›,๐›ผ) of p-valently star-like functions of order ๐›ผ if it satisfies the condition๎‚ตRe๐‘ง๐‘“๎…ž(๐‘ง)๎‚ถ๐‘“(๐‘ง)>๐›ผ(๐‘งโˆˆโ„‚;0โ‰ค๐›ผ<๐‘).(1.2) Furthermore, a function ๐‘“โˆˆ๐ด(๐‘,๐‘›) is said to be in the class ๐พ(๐‘,๐‘›,๐›ผ) of p-valently convex functions of order ๐›ผ if it satisfies the condition๎‚ตRe1+๐‘ง๐‘“๎…ž๎…ž(๐‘ง)๐‘“๎…ž๎‚ถ(๐‘ง)>๐›ผ(๐‘งโˆˆ๐‘ˆ;0โ‰ค๐›ผ<๐‘).(1.3) The classes ๐‘†(๐‘,๐‘›,๐›ผ) and ๐พ(๐‘,๐‘›,๐›ผ) were studied by Owa [1]. The class ๐‘†โˆ—(๐‘,๐›ผ)โˆถ=๐‘†(๐‘,1,๐›ผ) was considered by Patil and Thakare [2].

We denote by ๐‘‡(๐‘,๐‘›) the subclass of the class ๐ด(๐‘,๐‘›) consisting of functions of the form ๐‘“(๐‘ง)=๐‘ง๐‘โˆ’โˆž๎“๐‘˜=๐‘›๐‘Ž๐‘+๐‘˜๐‘ง๐‘+๐‘˜๎€ท๐‘Ž๐‘+๐‘˜๎€ธโ‰ฅ0;๐‘›,๐‘โˆˆโ„•(1.4) and define two further classes ๐‘‡โˆ—(๐‘,๐‘›,๐›ผ) and ๐ถ(๐‘,๐‘›,๐›ผ) by๐‘‡โˆ—(๐‘,๐‘›,๐›ผ)โˆถ=๐‘†(๐‘,๐‘›,๐›ผ)โˆฉ๐‘‡(๐‘,๐‘›),๐ถ(๐‘,๐‘›,๐›ผ)โˆถ=๐พ(๐‘,๐‘›,๐›ผ)โˆฉ๐‘‡(๐‘,๐‘›).(1.5) For the classes ๐‘‡โˆ—(๐‘,๐›ผ)โˆถ=๐‘†โˆ—(๐‘,๐›ผ)โˆฉ๐‘‡(๐‘),๐ถ(๐‘,๐›ผ)โˆถ=๐พ(๐‘,๐›ผ)โˆฉ๐‘‡(๐‘).

The following lemmas were given by Owa [3].

Lemma 1.1. Let the function ๐‘“ be defined by ๐‘“(๐‘ง)=๐‘ง๐‘โˆ’โˆž๎“๐‘˜=1๐‘Ž๐‘+๐‘˜๐‘ง๐‘+๐‘˜๎€ท๐‘Ž๐‘+๐‘˜๎€ธโ‰ฅ0;๐‘โˆˆโ„•.(1.6) Then, ๐‘“ is in the class ๐‘‡โˆ—(๐‘,๐›ผ) if and only if โˆž๎“๐‘˜=1(๐‘+๐‘˜โˆ’๐›ผ)๐‘Ž๐‘+๐‘˜โ‰ค๐‘โˆ’๐›ผ.(1.7) The result is sharp.

Lemma 1.2. Let the function ๐‘“ be defined by (1.6). Then, ๐‘“ is in the class ๐ถ(๐‘,๐›ผ) if and only if โˆž๎“๐‘˜=1(๐‘+๐‘˜)(๐‘+๐‘˜โˆ’๐›ผ)๐‘Ž๐‘+๐‘˜โ‰ค๐‘(๐‘โˆ’๐›ผ).(1.8) The result is sharp.

For a function ๐‘“ defined by (1.6) and in the class ๐‘‡โˆ—(๐‘,๐›ผ), Lemma 1.1 yields๐‘Ž๐‘+1โ‰ค๐‘โˆ’๐›ผ๐‘+1โˆ’๐›ผ.(1.9) On the other hand, for a function ๐‘“ defined by (1.6) and in the class ๐ถ(๐‘,๐›ผ), Lemma 1.2 yields๐‘Ž๐‘+1โ‰ค๐‘(๐‘โˆ’๐›ผ)(๐‘+1)(๐‘+1โˆ’๐›ผ).(1.10) In view of the coefficient inequalities (1.9) and (1.10), it would seem to be natural to introduce and study here two further classes ๐‘‡โˆ—๐œ€(๐‘,๐›ผ) and ๐ถ๐œ€(๐‘,๐›ผ) of analytic and p-valent functions, where ๐‘‡โˆ—๐œ€(๐‘,๐›ผ) denotes the subclass of ๐‘‡โˆ—(๐‘,๐›ผ) consisting of functions of the form๐‘“(๐‘ง)=๐‘ง๐‘โˆ’(๐‘โˆ’๐›ผ)๐œ€๐‘ง๐‘+1โˆ’๐›ผ๐‘+1โˆ’โˆž๎“๐‘˜=2๐‘Ž๐‘+๐‘˜๐‘ง๐‘+๐‘˜๎€ท๐‘Ž๐‘+๐‘˜๎€ธโ‰ฅ0,๐‘โˆˆโ„•,๐‘˜โˆˆโ„•โˆ’{1};0โ‰ค๐›ผ<๐‘;0โ‰ค๐œ€<1(1.11) and ๐ถ๐œ€(๐‘,๐›ผ) denotes the subclass of ๐ถ(๐‘,๐›ผ) consisting of functions of the form๐‘“(๐‘ง)=๐‘ง๐‘โˆ’๐‘(๐‘โˆ’๐›ผ)๐œ€๐‘ง(๐‘+1)(๐‘+1โˆ’๐›ผ)๐‘+1โˆ’โˆž๎“๐‘˜=2๐‘Ž๐‘+๐‘˜๐‘ง๐‘+๐‘˜๎€ท๐‘Ž๐‘+๐‘˜๎€ธ.โ‰ฅ0,๐‘โˆˆโ„•,๐‘˜โˆˆโ„•โˆ’{1};0โ‰ค๐›ผ<๐‘;0โ‰ค๐œ€<1(1.12) The classes ๐‘‡โˆ—๐œ€(๐‘,๐›ผ) and ๐ถ๐œ€(๐‘,๐›ผ) are studied by Aouf et al. [4].

The classes ๐‘‡โˆ—๐œ€(๐›ผ)โˆถ=๐‘‡โˆ—๐œ€(1,๐›ผ),๐ถ๐œ€(๐›ผ)โˆถ=๐ถ๐œ€(1,๐›ผ)(1.13)

were considered earlier by Silverman and Silvia [5].

Now, we give the following equalities for the functions ๐‘“(๐‘ง) belonging to the class ๐ด(๐‘,๐‘›):๐ท0๐ท๐‘“(๐‘ง)=๐‘“(๐‘ง),1๎€ท๐ท๐‘“(๐‘ง)=๐ท๐‘“(๐‘ง)=๐‘ง0๎€ธ๐‘“(๐‘ง)๎…ž๎ƒฌ=๐‘ง๐‘๐‘ง๐‘โˆ’1+โˆž๎“๐‘˜=๐‘›(๐‘+๐‘˜)๐‘Ž๐‘+๐‘˜๐‘ง๐‘+๐‘˜โˆ’1๎ƒญ=๐‘๐‘ง๐‘+โˆž๎“๐‘˜=๐‘›(๐‘+๐‘˜)๐‘Ž๐‘+๐‘˜๐‘ง๐‘+๐‘˜,๐ท2๐‘“๎€ท๐ท(๐‘ง)=๐ท(๐ท๐‘“(๐‘ง))=๐‘ง1๐‘“๎€ธ(๐‘ง)๎…ž๎ƒฌ=๐‘ง๐‘๐‘ง๐‘+โˆž๎“๐‘˜=๐‘›(๐‘+๐‘˜)๐‘Ž๐‘+๐‘˜๐‘ง๐‘+๐‘˜๎ƒญ๎…ž=๐‘2๐‘ง๐‘+โˆž๎“๐‘˜=๐‘›(๐‘+๐‘˜)2๐‘Ž๐‘+๐‘˜๐‘ง๐‘+๐‘˜,โ‹ฎ๐ทฮฉ๎€ท๐ท๐‘“(๐‘ง)=๐ทฮฉโˆ’1๎€ธ๐‘“(๐‘ง)=๐‘ฮฉ๐‘ง๐‘+โˆž๎“๐‘˜=๐‘›(๐‘˜+๐‘)ฮฉ๐‘Ž๐‘+๐‘˜๐‘ง๐‘+๐‘˜.(1.14) We define โ„˜โˆถ๐ด(๐‘,๐‘›)โ†’๐ด(๐‘,๐‘›) such that๎‚ต1โ„˜(ฮฉ,๐œ†,๐‘)=๐‘ฮฉ๎‚ถ๐ทโˆ’๐œ†ฮฉ๐œ†๐‘“(๐‘ง)+๐‘๐‘ง๎€ท๐ทฮฉ๎€ธ๐‘“(๐‘ง)๎…ž๎‚ต10โ‰ค๐œ†โ‰ค๐‘ฮฉ๎‚ถ,ฮฉโˆˆโ„•โˆช{0}.(1.15)

A function ๐‘“(๐‘ง)โˆˆ๐ด(๐‘,๐‘›) is said to be in the class โ„‘(ฮฉ,๐œ†,๐‘,๐›ผ) if it satisfies the inequality๎‚ปRe๐‘ง(โ„˜(ฮฉ,๐œ†,๐‘))โ€ฒ๎‚ผ๎ƒฏ๐‘ง๎€ทโ„˜(ฮฉ,๐œ†,๐‘)=Re1/๐‘ฮฉ๐ท+(1/๐‘โˆ’1)๐œ†๎€ธ๎€ทฮฉ๎€ธ๐‘“(๐‘ง)๎…ž๎€ท๐ท+(๐œ†/๐‘)๐‘งฮฉ๎€ธ๐‘“(๐‘ง)๎…ž๎…ž๎€ท1/๐‘ฮฉ๎€ธ๐ทโˆ’๐œ†ฮฉ๎€ท๐ท๐‘“(๐‘ง)+(๐œ†/๐‘)๐‘งฮฉ๎€ธ๐‘“(๐‘ง)๎…ž๎ƒฐ>๐›ผ,(1.16) for some ๐›ผ(0โ‰ค๐›ผ<๐‘),0โ‰ค๐œ†โ‰ค1/๐‘ฮฉ,โ€‰โ€‰ฮฉโˆˆโ„•โˆช{0} and for all ๐‘งโˆˆ๐‘ˆ.

If ฮฉ=0 and ๐œ†=0, we obtain the condition (1.2). Furthermore, we obtain the condition (1.3) for ฮฉ=0 and ๐œ†=1.

We denote by ๐‘‡(๐‘,๐‘›) the subclass of the class ๐ด(๐‘,๐‘›) consisting of functions of the form ๐‘“(๐‘ง)=๐‘ง๐‘โˆ’โˆž๎“๐‘˜=๐‘›๐‘Ž๐‘˜+๐‘๐‘ง๐‘˜+๐‘๎€ท๐‘Ž๐‘˜+๐‘๎€ธโ‰ฅ0;๐‘›,๐‘โˆˆโ„•,(1.17) and define the class โ„‘โˆ—(ฮฉ,๐œ†,๐‘,๐›ผ) byโ„‘โˆ—(ฮฉ,๐œ†,๐‘,๐›ผ)=โ„‘(ฮฉ,๐œ†,๐‘,๐›ผ)โˆฉ๐‘‡(๐‘,๐‘›).(1.18)

Furthermore, we denote by โ„‘โˆ—๐œ€(ฮฉ,๐œ†,๐‘,๐›ผ) the subclass of โ„‘โˆ—(ฮฉ,๐œ†,๐‘,๐›ผ) consisting of functions of the form ๐‘“(๐‘ง)=๐‘ง๐‘โˆ’(๐‘โˆ’๐›ผ)๐œ€((๐‘+๐‘›)/๐‘)ฮฉ๎€ท1+๐œ†๐‘›๐‘ฮฉโˆ’1๎€ธ๐‘ง(๐‘›+๐‘โˆ’๐›ผ)๐‘+๐‘›โˆ’โˆž๎“๐‘˜=๐‘›+1๐‘Ž๐‘+๐‘˜๐‘ง๐‘+๐‘˜(0โ‰ค๐œ€<1).(1.19) The main object of the present paper is to investigate interesting properties and characteristics of the classes โ„‘โˆ—(ฮฉ,๐œ†,๐‘,๐›ผ) and โ„‘โˆ—๐œ€(ฮฉ,๐œ†,๐‘,๐›ผ). Also, the partial sums is defined for ๐‘“ function defined by (1.19).

2. A Coefficient Inequality for the Class โ„‘โˆ—(ฮฉ,๐œ†,๐‘,๐›ผ) and Some Theorems for the Class โ„‘โˆ—๐œ€(ฮฉ,๐œ†,๐‘,๐›ผ)

First, we give a coefficient inequality for the class โ„‘โˆ—(ฮฉ,๐œ†,๐‘,๐›ผ).

Theorem 2.1. Let the function ๐‘“ be defined by (1.17). Then, ๐‘“ is in the class โ„‘โˆ—(ฮฉ,๐œ†,๐‘,๐›ผ) if and only if โˆž๎“๐‘˜=n๎‚ต๐‘˜+๐‘๐‘๎‚ถฮฉ๎€ท1+๐œ†๐‘˜๐‘ฮฉโˆ’1๎€ธ(๐‘˜+๐‘โˆ’๐›ผ)๐‘Ž๐‘+๐‘˜โ‰ค๐‘โˆ’๐›ผ.(2.1)

Proof. Suppose that ๐‘“(๐‘ง)โˆˆโ„‘โˆ—(ฮฉ,๐œ†,๐‘,๐›ผ). Then, we find from (1.16) that ๎‚ปRe๐‘ง(โ„˜(ฮฉ,๐œ†,๐‘))๎…ž๎‚ผ๎ƒฏ๐‘ง๎€ทโ„˜(ฮฉ,๐œ†,๐‘)=Re1/๐‘ฮฉ๐ท+(1/๐‘โˆ’1)๐œ†๎€ธ๎€ทฮฉ๎€ธ๐‘“(๐‘ง)๎…ž๎€ท๐ท+(๐œ†/๐‘)๐‘งฮฉ๎€ธ๐‘“(๐‘ง)๎…ž๎…ž๎€ท1/๐‘ฮฉ๎€ธ๐ทโˆ’๐œ†ฮฉ๎€ท๐ท๐‘“(๐‘ง)+(๐œ†/๐‘)๐‘งฮฉ๎€ธ๐‘“(๐‘ง)๎…ž๎ƒฐ๎ƒฏ=Re๐‘๐‘ง๐‘โˆ’โˆ‘โˆž๐‘˜=๐‘›(๐‘˜+๐‘)ฮฉ+1๎€ท1/๐‘ฮฉ๎€ธ๐‘Ž+๐œ†๐‘˜/๐‘๐‘+๐‘˜๐‘ง๐‘+๐‘˜๐‘ง๐‘โˆ’โˆ‘โˆž๐‘˜=๐‘›(๐‘˜+๐‘)ฮฉ๎€ท1/๐‘ฮฉ๎€ธ๐‘Ž+๐œ†๐‘˜/๐‘๐‘+๐‘˜๐‘ง๐‘+๐‘˜๎ƒฐ>๐›ผ.(2.2) If we choose ๐‘ง to be real and let ๐‘งโ†’1โˆ’, we get ๎ƒฏโˆ‘๐‘โˆ’โˆž๐‘˜=๐‘›(๐‘˜+๐‘)ฮฉ+1๎€ท1/๐‘ฮฉ๎€ธ๐‘Ž+๐œ†๐‘˜/๐‘๐‘+๐‘˜โˆ‘1โˆ’โˆž๐‘˜=๐‘›(๐‘˜+๐‘)ฮฉ๎€ท1/๐‘ฮฉ๎€ธ๐‘Ž+๐œ†๐‘˜/๐‘๐‘+๐‘˜๎ƒฐโ‰ฅ๐›ผ(2.3) or, equivalently, ๐‘โˆ’โˆž๎“๐‘˜=๐‘›(๐‘˜+๐‘)ฮฉ+1๎€ท1/๐‘ฮฉ๎€ธ๐‘Ž+๐œ†๐‘˜/๐‘๐‘+๐‘˜๎ƒฏโ‰ฅ๐›ผ1โˆ’โˆž๎“๐‘˜=๐‘›(๐‘˜+๐‘)ฮฉ๎€ท1/๐‘ฮฉ๎€ธ๐‘Ž+๐œ†๐‘˜/๐‘๐‘+๐‘˜๎ƒฐ.(2.4) Thus, we have โˆž๎“๐‘˜=๐‘›๎€ฝ(๐‘˜+๐‘)ฮฉ+1๎€ท1/๐‘ฮฉ๎€ธ+๐œ†๐‘˜/๐‘โˆ’(๐‘˜+๐‘)ฮฉ๎€ท1/๐‘ฮฉ๐‘Ž+๐œ†๐‘˜/๐‘๎€ธ๎€พ๐‘+๐‘˜โ‰ค๐‘โˆ’๐›ผ(2.5) or โˆž๎“๐‘˜=๐‘›๎ƒฏ๎‚ต๐‘˜+๐‘๐‘๎‚ถฮฉ๎€ท1+๐œ†๐‘˜๐‘ฮฉโˆ’1๎€ธ๎ƒฐ๐‘Ž(๐‘˜+๐‘โˆ’๐›ผ)๐‘+๐‘˜โ‰ค๐‘โˆ’๐›ผ.(2.6) Conversely, suppose that the inequality (2.1) holds true and let ๐‘งโˆˆ๐œ•๐‘ˆ={๐‘งโˆถ๐‘งโˆˆโ„‚,|๐‘ง|=1}.(2.7) Then, we find from the definition (1.4) that ||||๐‘ง(โ„˜(ฮฉ,๐œ†,๐‘))๎…ž||||=|||||๐‘ง๎€ทโ„˜(ฮฉ,๐œ†,๐‘)โˆ’๐‘1/๐‘ฮฉ๐ท+(1/๐‘โˆ’1)๐œ†๎€ธ๎€ทฮฉ๎€ธ๐‘“(๐‘ง)๎…ž๎€ท๐ท+(๐œ†/๐‘)๐‘งฮฉ๎€ธ๐‘“(๐‘ง)๎…ž๎…ž๎€ท1/๐‘ฮฉ๎€ธ๐ทโˆ’๐œ†ฮฉ๎€ท๐ท๐‘“(๐‘ง)+(๐œ†/๐‘)๐‘งฮฉ๎€ธ๐‘“(๐‘ง)๎…ž|||||=||||โˆ’๐‘๐‘๐‘ง๐‘โˆ’โˆ‘โˆž๐‘˜=๐‘›((๐‘˜+๐‘)/๐‘)ฮฉ๎€ท(๐‘˜+๐‘)1+๐œ†๐‘˜๐‘ฮฉโˆ’1๎€ธ๐‘Ž๐‘+๐‘˜๐‘ง๐‘+๐‘˜๐‘ง๐‘โˆ’โˆ‘โˆž๐‘˜=๐‘›((๐‘˜+๐‘)/๐‘)ฮฉ๎€ท1+๐œ†๐‘˜๐‘ฮฉโˆ’1๎€ธ๐‘Ž๐‘+๐‘˜๐‘ง๐‘+๐‘˜||||โ‰คโˆ‘โˆ’๐‘โˆž๐‘˜=๐‘›๐‘˜((๐‘˜+๐‘)/๐‘)ฮฉ๎€ท1+๐œ†๐‘˜๐‘ฮฉโˆ’1๎€ธ๐‘Ž๐‘+๐‘˜โˆ‘1โˆ’โˆž๐‘˜=๐‘›((๐‘˜+๐‘)/๐‘)ฮฉ๎€ท1+๐œ†๐‘˜๐‘ฮฉโˆ’1๎€ธ๐‘Ž๐‘+๐‘˜.(2.8) By means of inequality (2.1), we can write โˆž๎“๐‘˜=๐‘›๎‚ต๐‘˜+๐‘๐‘๎‚ถฮฉ๎€ท1+๐œ†๐‘˜๐‘ฮฉโˆ’1๎€ธ(๐‘˜+๐‘โˆ’๐›ผ)๐‘Ž๐‘+๐‘˜โ‰ค๐‘โˆ’๐›ผโŸนโˆž๎“๐‘˜=๐‘›๐‘˜๎‚ต๐‘˜+๐‘๐‘๎‚ถฮฉ๎€ท1+๐œ†๐‘˜๐‘ฮฉโˆ’1๎€ธ๐‘Ž๐‘+๐‘˜โ‰ค๐‘โˆ’๐›ผโˆ’โˆž๎“๐‘˜=๐‘›๎‚ต๐‘˜+๐‘๐‘๎‚ถฮฉ๎€ท1+๐œ†๐‘˜๐‘ฮฉโˆ’1๎€ธ(๐‘โˆ’๐›ผ)๐‘Ž๐‘+๐‘˜(2.9) or โˆž๎“๐‘˜=๐‘›๐‘˜๎‚ต๐‘˜+๐‘๐‘๎‚ถฮฉ๎€ท1+๐œ†๐‘˜๐‘ฮฉโˆ’1๎€ธ๐‘Ž๐‘+๐‘˜โ‰ค๐‘โˆ’๐›ผโˆ’(๐‘โˆ’๐›ผ)โˆž๎“๐‘˜=๐‘›๎‚ต๐‘˜+๐‘๐‘๎‚ถฮฉ๎€ท1+๐œ†๐‘˜๐‘ฮฉโˆ’1๎€ธ๐‘Ž๐‘+๐‘˜.(2.10) Thus, we obtain |||||๐‘ง๎€ท1/๐‘ฮฉ๐ท+(1/๐‘โˆ’1)๐œ†๎€ธ๎€ทฮฉ๎€ธ๐‘“(๐‘ง)๎…ž๎€ท๐ท+(๐œ†/๐‘)๐‘งฮฉ๎€ธ๐‘“(๐‘ง)๎…ž๎…ž๎€ท1/๐‘ฮฉ๎€ธ๐ทโˆ’๐œ†ฮฉ๎€ท๐ท๐‘“(๐‘ง)+(๐œ†/๐‘)๐‘งฮฉ๎€ธ๐‘“(๐‘ง)๎…ž|||||=||||โˆ‘โˆ’๐‘โˆž๐‘˜=๐‘›๐‘˜((๐‘˜+๐‘)/๐‘)ฮฉ๎€ท1+๐œ†๐‘˜๐‘ฮฉโˆ’1๎€ธ๐‘Ž๐‘+๐‘˜๐‘ง๐‘˜โˆ‘1โˆ’โˆž๐‘˜=๐‘›((๐‘˜+๐‘)/๐‘)ฮฉ๎€ท1+๐œ†๐‘˜๐‘ฮฉโˆ’1๎€ธ๐‘Ž๐‘+๐‘˜๐‘ง๐‘˜||||โ‰คโˆ‘๐‘โˆ’๐›ผโˆ’(๐‘โˆ’๐›ผ)โˆž๐‘˜=๐‘›((๐‘˜+๐‘)/๐‘)ฮฉ๎€ท1+๐œ†๐‘˜๐‘ฮฉโˆ’1๎€ธ๐‘Ž๐‘+๐‘˜โˆ‘1โˆ’โˆž๐‘˜=๐‘›((๐‘˜+๐‘)/๐‘)ฮฉ๎€ท1+๐œ†๐‘˜๐‘ฮฉโˆ’1๎€ธ๐‘Ž๐‘+๐‘˜=๐‘โˆ’๐›ผ.(2.11) This evidently completes the proof of Theorem 2.1.

Now, we give a characterization theorem for the class โ„‘โˆ—๐œ€(ฮฉ,๐œ†,๐‘,๐›ผ).

Theorem 2.2. Let the function ๐‘“ be defined by (1.19). Then, ๐‘“ is in the class โ„‘โˆ—๐œ€(ฮฉ,๐œ†,๐‘,๐›ผ) if and only if โˆž๎“๐‘˜=๐‘›+1๎‚ต๐‘˜+๐‘๐‘๎‚ถฮฉ๎€ท1+๐œ†๐‘˜๐‘ฮฉโˆ’1๎€ธ(๐‘˜+๐‘โˆ’๐›ผ)๐‘Ž๐‘+๐‘˜๎‚ต1โ‰ค(๐‘โˆ’๐›ผ)(1โˆ’๐œ€)0โ‰ค๐›ผ<๐‘,0โ‰ค๐œ€<1,0โ‰ค๐œ†โ‰ค๐‘ฮฉ๎‚ถ.(2.12) The result is sharp for the function ๐‘“ given by ๐‘“(๐‘ง)=๐‘ง๐‘โˆ’(๐‘โˆ’๐›ผ)๐œ€((๐‘+๐‘›)/๐‘)ฮฉ๎€ท1+๐œ†๐‘›๐‘ฮฉโˆ’1๎€ธ๐‘ง(๐‘›+๐‘โˆ’๐›ผ)๐‘+๐‘›โˆ’(๐‘โˆ’๐›ผ)(1โˆ’๐œ€)((๐‘+๐‘˜)/๐‘)ฮฉ๎€ท1+๐œ†๐‘˜๐‘ฮฉโˆ’1๎€ธ(๐‘ง๐‘˜+๐‘โˆ’๐›ผ)๐‘+๐‘˜(๐‘˜=๐‘›+1,๐‘›+2,โ€ฆ;๐‘›โˆˆโ„•).(2.13)

Proof. Using inequality (2.1), we have ๎‚ต๐‘›+๐‘๐‘๎‚ถฮฉ๎€ท1+๐œ†๐‘›๐‘ฮฉโˆ’1๎€ธ(๐‘›+๐‘โˆ’๐›ผ)๐‘Ž๐‘+๐‘›+โˆž๎“๐‘˜=๐‘›+1๎‚ต๐‘˜+๐‘๐‘๎‚ถฮฉ๎€ท1+๐œ†๐‘˜๐‘ฮฉโˆ’1๎€ธ(๐‘˜+๐‘โˆ’๐›ผ)๐‘Ž๐‘+๐‘˜โ‰ค๐‘โˆ’๐›ผ(2.14) or โˆž๎“๐‘˜=๐‘›+1๎‚ต๐‘˜+๐‘๐‘๎‚ถฮฉ๎€ท1+๐œ†๐‘˜๐‘ฮฉโˆ’1๎€ธ(๐‘˜+๐‘โˆ’๐›ผ)๐‘Ž๐‘+๐‘˜๎‚ตโ‰ค๐‘โˆ’๐›ผโˆ’๐‘›+๐‘๐‘๎‚ถฮฉ๎€ท1+๐œ†๐‘›๐‘ฮฉโˆ’1๎€ธ(๐‘›+๐‘โˆ’๐›ผ)๐‘Ž๐‘+๐‘›.(2.15) Thus, by setting ๐‘Ž๐‘+๐‘›=(๐‘โˆ’๐›ผ)๐œ€((๐‘+๐‘›)/๐‘)ฮฉ๎€ท1+๐œ†๐‘›๐‘ฮฉโˆ’1๎€ธ(๐‘›+๐‘โˆ’๐›ผ),(2.16) we obtain โˆž๎“๐‘˜=๐‘›+1๎‚ต๐‘˜+๐‘๐‘๎‚ถฮฉ๎€ท1+๐œ†๐‘˜๐‘ฮฉโˆ’1๎€ธ(๐‘˜+๐‘โˆ’๐›ผ)๐‘Ž๐‘+๐‘˜๎‚ตโ‰ค๐‘โˆ’๐›ผโˆ’๐‘›+๐‘๐‘๎‚ถฮฉ๎€ท1+๐œ†๐‘›๐‘ฮฉโˆ’1๎€ธ(๐‘›+๐‘โˆ’๐›ผ)(๐‘โˆ’๐›ผ)๐œ€((๐‘+๐‘›)/๐‘)ฮฉ๎€ท1+๐œ†๐‘›๐‘ฮฉโˆ’1๎€ธ(๐‘›+๐‘โˆ’๐›ผ)(2.17) or โˆž๎“๐‘˜=๐‘›+1๎‚ต๐‘˜+๐‘๐‘๎‚ถฮฉ๎€ท1+๐œ†๐‘˜๐‘ฮฉโˆ’1๎€ธ(๐‘˜+๐‘โˆ’๐›ผ)๐‘Ž๐‘+๐‘˜โ‰ค(pโˆ’๐›ผ)(1โˆ’๐œ€).(2.18) A closure theorem for the class โ„‘โˆ—๐œ€(ฮฉ,๐œ†,๐‘,๐›ผ) is given by the following.

Theorem 2.3. Let ๐‘“๐‘ (๐‘ง)=๐‘ง๐‘โˆ’(๐‘โˆ’๐›ผ)๐œ€((๐‘+๐‘›)/๐‘)ฮฉ๎€ท1+๐œ†๐‘›๐‘ฮฉโˆ’1๎€ธ๐‘ง(๐‘›+๐‘โˆ’๐›ผ)๐‘+๐‘›โˆ’โˆž๎“๐‘˜=๐‘›+1๐‘Ž๐‘+๐‘˜,๐‘ ๐‘ง๐‘+๐‘˜๎€ท๐‘Ž๐‘+๐‘˜,๐‘ ๎€ธ.โ‰ฅ0;๐‘,๐‘›โˆˆโ„•;0โ‰ค๐œ€<1;๐‘ =1,2,3,โ€ฆ,๐‘š(2.19) If ๐‘“๐‘ โˆˆโ„‘โˆ—๐œ€(ฮฉ,๐œ†,๐‘,๐›ผ), then the function ๐‘” given by ๐‘”(๐‘ง)=๐‘ง๐‘โˆ’(๐‘โˆ’๐›ผ)๐œ€((๐‘+๐‘›)/๐‘)ฮฉ๎€ท1+๐œ†๐‘›๐‘ฮฉโˆ’1๎€ธ๐‘ง(๐‘›+๐‘โˆ’๐›ผ)๐‘+๐‘›โˆ’โˆž๎“k=๐‘›+1๐‘๐‘+๐‘˜๐‘ง๐‘+๐‘˜,(2.20) with ๐‘๐‘+๐‘˜1โˆถ=๐‘š๐‘š๎“๐‘ =1๐‘Ž๐‘+๐‘˜,๐‘ โ‰ฅ0,(2.21) is also in the class โ„‘โˆ—๐œ€(ฮฉ,๐œ†,๐‘,๐›ผ).

Proof. Since ๐‘“๐‘ โˆˆโ„‘โˆ—๐œ€(ฮฉ,๐œ†,๐‘,๐›ผ) for ๐‘ =1,2,โ€ฆ,๐‘š, it follows from Theorem 2.2 that โˆž๎“๐‘˜=๐‘›+1๎‚ต๐‘˜+๐‘๐‘๎‚ถฮฉ๎€ท1+๐œ†๐‘˜๐‘ฮฉโˆ’1๎€ธ(๐‘˜+๐‘โˆ’๐›ผ)๐‘Ž๐‘+๐‘˜,๐‘ โ‰ค(๐‘โˆ’๐›ผ)(1โˆ’๐œ€)(๐‘ =1,2,โ€ฆ,๐‘š).(2.22) By applying (2.22) and the definition (2.21), we write โˆž๎“๐‘˜=๐‘›+1๎‚ต๐‘˜+๐‘๐‘๎‚ถฮฉ๎€ท1+๐œ†๐‘˜๐‘ฮฉโˆ’1๎€ธ(๐‘˜+๐‘โˆ’๐›ผ)๐‘๐‘+๐‘˜=โˆž๎“๐‘˜=๐‘›+1๎‚ต๐‘˜+๐‘๐‘๎‚ถฮฉ๎€ท1+๐œ†๐‘˜๐‘ฮฉโˆ’1๎€ธ(๎ƒฏ1๐‘˜+๐‘โˆ’๐›ผ)๐‘š๐‘š๎“๐‘ =1๐‘Ž๐‘+๐‘˜,๐‘ ๎ƒฐ=1๐‘š๐‘š๎“๐‘ =1โŽงโŽชโŽจโŽชโŽฉโˆž๎“๐‘˜=๐‘›+1๎‚ต๐‘˜+๐‘๐‘๎‚ถฮฉ๎€ท1+๐œ†๐‘˜๐‘ฮฉโˆ’1๎€ธ(๐‘˜+๐‘โˆ’๐›ผ)๐‘Ž๐‘+๐‘˜,๐‘ โŽซโŽชโŽฌโŽชโŽญโ‰ค1๐‘š๐‘š๎“๐‘ =1(๐‘โˆ’๐›ผ)(1โˆ’๐œ€)=(๐‘โˆ’๐›ผ)(1โˆ’๐œ€).(2.23)

Theorem 2.4. Let ๐‘“๐‘+๐‘›(๐‘ง)=๐‘ง๐‘โˆ’(๐‘โˆ’๐›ผ)๐œ€((๐‘+๐‘›)/๐‘)ฮฉ๎€ท1+๐œ†๐‘›๐‘ฮฉโˆ’1๎€ธ๐‘ง(๐‘›+๐‘โˆ’๐›ผ)๐‘+๐‘›,๐‘“๐‘+๐‘˜(๐‘ง)=๐‘ง๐‘โˆ’(๐‘โˆ’๐›ผ)๐œ€((๐‘+๐‘›)/๐‘)ฮฉ๎€ท1+๐œ†๐‘›๐‘ฮฉโˆ’1๎€ธ(๐‘ง๐‘›+๐‘โˆ’๐›ผ)๐‘+๐‘›โˆ’(๐‘โˆ’๐›ผ)(1โˆ’๐œ€)((๐‘+๐‘˜)/๐‘)ฮฉ๎€ท1+๐œ†๐‘˜๐‘ฮฉโˆ’1๎€ธ๐‘ง(๐‘˜+๐‘โˆ’๐›ผ)๐‘+๐‘˜,(2.24) where ๐‘˜โˆˆ{๐‘›+1,๐‘›+2,โ€ฆ}.
Then, ๐‘“ is in the class โ„‘โˆ—๐œ€(ฮฉ,๐œ†,๐‘,๐›ผ) if and only if it can be expressed in the form ๐‘“(๐‘ง)=โˆž๎“๐‘˜=๐‘›๐œ‚๐‘+๐‘˜๐‘“๐‘+๐‘˜๎ƒฉ๐œ‚(๐‘ง)๐‘+๐‘˜โ‰ฅ0;โˆž๎“๐‘˜=๐‘›๐œ‚๐‘+๐‘˜๎ƒช=1.(2.25)

Proof. Suppose that ๐‘“ is given by (2.25), so that we find from (2.24) that ๐‘“(๐‘ง)=๐‘ง๐‘โˆ’(๐‘โˆ’๐›ผ)๐œ€((๐‘+๐‘›)/๐‘)ฮฉ๎€ท1+๐œ†๐‘›๐‘ฮฉโˆ’1๎€ธ๐‘ง(๐‘›+๐‘โˆ’๐›ผ)๐‘+๐‘›โˆ’โˆž๎“๐‘˜=๐‘›+1(๐‘โˆ’๐›ผ)(1โˆ’๐œ€)((๐‘+๐‘˜)/๐‘)ฮฉ๎€ท1+๐œ†๐‘˜๐‘ฮฉโˆ’1๎€ธ(๐œ‚๐‘˜+๐‘โˆ’๐›ผ)๐‘+๐‘˜๐‘ง๐‘+๐‘˜,(2.26) where the coefficients ๐œ‚๐‘+๐‘˜ are given with โˆ‘โˆž๐‘˜=๐‘›๐œ‚๐‘+๐‘˜=1,โ€‰โ€‰๐œ‚๐‘+๐‘˜โ‰ฅ0. Then, since, โˆž๎“๐‘˜=๐‘›+1๎‚ต๐‘˜+๐‘๐‘๎‚ถฮฉ๎€ท1+๐œ†๐‘˜๐‘ฮฉโˆ’1๎€ธ(๐‘˜+๐‘โˆ’๐›ผ)(๐‘โˆ’๐›ผ)(1โˆ’๐œ€)((๐‘˜+๐‘)/๐‘)ฮฉ๎€ท1+๐œ†๐‘˜๐‘ฮฉโˆ’1๎€ธ๐œ‚(๐‘˜+๐‘โˆ’๐›ผ)๐‘+๐‘˜=(๐‘โˆ’๐›ผ)(1โˆ’๐œ€)โˆž๎“๐‘˜=๐‘›+1๐œ‚๐‘+๐‘˜๎€ท=(๐‘โˆ’๐›ผ)(1โˆ’๐œ€)1โˆ’๐œ‚๐‘+๐‘›๎€ธโ‰ค(๐‘โˆ’๐›ผ)(1โˆ’๐œ€),(2.27) we conclude from Theorem 2.2 that ๐‘“โˆˆโ„‘โˆ—๐œ€(ฮฉ,๐œ†,๐‘,๐›ผ).
Conversely, let us assume that the function ๐‘“ defined by (1.19) is in the class โ„‘โˆ—๐œ€(ฮฉ,๐œ†,๐‘,๐›ผ). Then, ๐‘Ž๐‘+๐‘˜โ‰ค(๐‘โˆ’๐›ผ)(1โˆ’๐œ€)((๐‘˜+๐‘)/๐‘)ฮฉ๎€ท1+๐œ†๐‘˜๐‘ฮฉโˆ’1๎€ธ(๐‘˜+๐‘โˆ’๐›ผ),(2.28) which follows readily from (2.12) for ๐‘˜โˆˆ{๐‘›+1,๐‘›+2,โ€ฆ}.
For ๐‘˜โˆˆ{๐‘›+1,๐‘›+2,โ€ฆ}, setting ๐œ‚๐‘+๐‘˜=((๐‘˜+๐‘)/๐‘)ฮฉ๎€ท1+๐œ†๐‘˜๐‘ฮฉโˆ’1๎€ธ(๐‘˜+๐‘โˆ’๐›ผ)๐‘Ž(๐‘โˆ’๐›ผ)(1โˆ’๐œ€)๐‘+๐‘˜(2.29) and ๐œ‚๐‘+๐‘›โˆ‘=1โˆ’โˆž๐‘˜=๐‘›+1๐œ‚๐‘+๐‘˜, we thus arrive at (2.25). This completes the proof of Theorem 2.4.

Theorem 2.5. Let ๐‘“ be given by (1.19) and define the partial sums ๐‘ ๐‘š(๐‘ง) by ๐‘ ๐‘šโŽงโŽชโŽชโŽจโŽชโŽชโŽฉ๐‘ง(๐‘ง)=๐‘โˆ’๐‘ฮฉ(๐‘โˆ’๐›ผ)๐œ€(๐‘+๐‘›)ฮฉ๎€ท1+๐œ†๐‘›๐‘ฮฉโˆ’1๎€ธ๐‘ง(๐‘›+๐‘โˆ’๐›ผ)๐‘+๐‘›๐‘ง,๐‘š=๐‘›,๐‘โˆ’๐‘ฮฉ(๐‘โˆ’๐›ผ)๐œ€(๐‘+๐‘›)ฮฉ๎€ท1+๐œ†๐‘›๐‘ฮฉโˆ’1๎€ธ(๐‘ง๐‘›+๐‘โˆ’๐›ผ)๐‘+๐‘›โˆ’๐‘š๎“๐‘˜=๐‘›+1๐‘Ž๐‘+๐‘˜๐‘ง๐‘+๐‘˜,๐‘š=๐‘›+1,๐‘›+2,โ€ฆ.(2.30) Suppose also that โˆž๎“๐‘˜=๐‘›+1๐‘‘๐‘+๐‘˜๐‘Ž๐‘+๐‘˜๐‘โ‰ค1โˆ’ฮฉ(๐‘โˆ’๐›ผ)๐œ€(๐‘+๐‘›)ฮฉ๎€ท1+๐œ†๐‘›๐‘ฮฉโˆ’1๎€ธ,๎ƒฉ(๐‘›+๐‘โˆ’๐›ผ)where๐‘‘๐‘+๐‘˜=(๐‘+๐‘˜)ฮฉ๎€ท1+๐œ†๐‘˜๐‘ฮฉโˆ’1๎€ธ(๐‘˜+๐‘โˆ’๐›ผ)๐‘ฮฉ1(๐‘โˆ’๐›ผ)(1โˆ’๐œ€);0โ‰ค๐›ผ<๐‘,0โ‰ค๐œ€<1,0โ‰ค๐œ†โ‰ค๐‘ฮฉ๎ƒช.(2.31) Then, for ๐‘šโ‰ฅ๐‘›+1, one has ๎‚ปRe๐‘“(๐‘ง)๐‘ ๐‘š๎‚ผ๐‘(๐‘ง)>1โˆ’ฮฉ(๐‘โˆ’๐›ผ)(1โˆ’๐œ€)(๐‘+๐‘š+1)ฮฉ๎€บ1+๐œ†(๐‘š+1)๐‘ฮฉโˆ’1๎€ป,๎‚ต๐‘ (๐‘+๐‘š+1โˆ’๐›ผ)(2.32)Re๐‘š(๐‘ง)๎‚ถ>๐‘“(๐‘ง)(๐‘+๐‘š+1)ฮฉ๎€ฝ1+๐œ†(๐‘š+1)๐‘ฮฉโˆ’1๎€พ(๐‘š+๐‘+1โˆ’๐›ผ)๐‘ฮฉ(๐‘โˆ’๐›ผ)(1โˆ’๐œ€)+(๐‘+๐‘š+1)ฮฉ๎€ฝ1+๐œ†(๐‘š+1)๐‘ฮฉโˆ’1๎€พ(๐‘š+๐‘+1โˆ’๐›ผ).(2.33) Each of the bounds in (2.32) and (2.33) is the best possible.

Proof. From (2.31) and Theorem 2.2, we have that ๐‘“โˆˆโ„‘โˆ—๐œ€(ฮฉ,๐œ†,๐‘,๐›ผ). By definition ๐‘‘๐‘+๐‘˜, we can write ๐‘‘๐‘+๐‘˜=((๐‘+๐‘˜)/๐‘)ฮฉ๎€ท1+๐œ†๐‘˜๐‘ฮฉโˆ’1๎€ธ(๐‘˜+๐‘โˆ’๐›ผ)=๎‚ต๐‘˜(๐‘โˆ’๐›ผ)(1โˆ’๐œ€)1+๐‘๎‚ถฮฉ๎€ท1+๐œ†๐‘˜๐‘ฮฉโˆ’1๎€ธ๎‚ธ๐‘โˆ’๐›ผ+๐‘˜๎‚น=๎‚ต๐‘˜(๐‘โˆ’๐›ผ)(1โˆ’๐œ€)1+๐‘๎‚ถฮฉ๎€ท1+๐œ†๐‘˜๐‘ฮฉโˆ’1๎€ธ๎‚ธ1+๐‘˜1โˆ’๐œ€๎‚น(๐‘โˆ’๐›ผ)(1โˆ’๐œ€)>1.(2.34) Under the hypothesis of this theorem, we can see from (2.31) that ๐‘‘๐‘+๐‘˜+1>๐‘‘๐‘+๐‘˜>1, for ๐‘˜=๐‘›+1,๐‘›+2โ€ฆ. Therefore, we have ๐‘š๎“๐‘˜=๐‘›+1๐‘Ž๐‘+๐‘˜+๐‘‘โˆž๐‘+๐‘š+1๎“๐‘˜=๐‘š+1๐‘Ž๐‘+๐‘˜โ‰คโˆž๎“๐‘˜=๐‘›+1๐‘‘๐‘+๐‘˜๐‘Ž๐‘+๐‘˜๐‘โ‰ค1โˆ’ฮฉ(๐‘โˆ’๐›ผ)๐œ€(๐‘+๐‘›)ฮฉ๎€ท1+๐œ†๐‘›๐‘ฮฉโˆ’1๎€ธ(๐‘+๐‘›โˆ’๐›ผ).(2.35) Using (1.19) and (2.30), we can write ๐‘“(๐‘ง)๐‘ ๐‘š=๐‘ง(๐‘ง)๐‘โˆ’๎€ท๐‘ฮฉ(๐‘โˆ’๐›ผ)๐œ€/(๐‘+๐‘›)ฮฉ๎€ท1+๐œ†๐‘›๐‘ฮฉโˆ’1๎€ธ๎€ธ๐‘ง(๐‘›+๐‘โˆ’๐›ผ)๐‘+๐‘›โˆ’โˆ‘โˆž๐‘˜=๐‘›+1๐‘Ž๐‘+๐‘˜๐‘ง๐‘+๐‘˜๐‘ง๐‘โˆ’๎€ท๐‘ฮฉ(๐‘โˆ’๐›ผ)๐œ€/(๐‘+๐‘›)ฮฉ๎€ท1+๐œ†๐‘›๐‘ฮฉโˆ’1๎€ธ๎€ธ๐‘ง(๐‘›+๐‘โˆ’๐›ผ)๐‘+๐‘›โˆ’โˆ‘๐‘š๐‘˜=๐‘›+1๐‘Ž๐‘+๐‘˜๐‘ง๐‘+๐‘˜โˆ‘=1โˆ’โˆž๐‘˜=๐‘š+1๐‘Ž๐‘+๐‘˜๐‘ง๐‘+๐‘˜๐‘ง๐‘โˆ’๎€ท๐‘ฮฉ(๐‘โˆ’๐›ผ)๐œ€/(๐‘+๐‘›)ฮฉ๎€ท1+๐œ†๐‘›๐‘ฮฉโˆ’1๎€ธ๎€ธ๐‘ง(๐‘›+๐‘โˆ’๐›ผ)๐‘+๐‘›โˆ’โˆ‘๐‘š๐‘˜=๐‘›+1๐‘Ž๐‘+๐‘˜๐‘ง๐‘+๐‘˜.(2.36) Set ฮจ1(๐‘ง)=๐‘‘๐‘+๐‘š+1๎‚ธ๐‘“(๐‘ง)๐‘ ๐‘šโˆ’๎‚ต1(๐‘ง)1โˆ’๐‘‘๐‘+๐‘š+1๐‘‘๎‚ถ๎‚น=1โˆ’๐‘+๐‘š+1โˆ‘โˆž๐‘˜=๐‘š+1๐‘Ž๐‘+๐‘˜๐‘ง๐‘˜๎€ท๐‘1โˆ’ฮฉ(๐‘โˆ’๐›ผ)๐œ€/(๐‘+๐‘›)ฮฉ๎€ท1+๐œ†๐‘›๐‘ฮฉโˆ’1๎€ธ๎€ธ๐‘ง(๐‘›+๐‘โˆ’๐›ผ)๐‘›โˆ’โˆ‘๐‘š๐‘˜=๐‘›+1๐‘Ž๐‘+๐‘˜๐‘ง๐‘˜.(2.37) By applying (2.35) and (2.37), we find that ||||๐œ“1(๐‘ง)โˆ’1๐œ“1||||=||||||||||(๐‘ง)+1โˆ’๐‘‘๐‘+๐‘š+1โˆ‘โˆž๐‘˜=๐‘š+1๐‘Ž๐‘+๐‘˜๐‘ง๐‘˜๎ƒฉ2โˆ’2๐‘ฮฉ(๐‘โˆ’๐›ผ)๐œ€(๐‘+๐‘›)ฮฉ๎€ท1+๐œ†๐‘›๐‘ฮฉโˆ’1๎€ธ๎ƒช๐‘ง(๐‘›+๐‘โˆ’๐›ผ)๐‘›โˆ‘โˆ’2๐‘š๐‘˜=๐‘›+1๐‘Ž๐‘+๐‘˜๐‘ง๐‘˜โˆ’๐‘‘๐‘+๐‘š+1โˆ‘โˆž๐‘˜=๐‘š+1๐‘Ž๐‘+๐‘˜๐‘ง๐‘˜||||||||||โ‰ค๐‘‘๐‘+๐‘š+1โˆ‘โˆž๐‘˜=๐‘š+1๐‘Ž๐‘+๐‘˜๐‘ง๐‘+๐‘˜2๎ƒฌ๎ƒฉ๐‘1โˆ’ฮฉ(๐‘โˆ’๐›ผ)๐œ€(๐‘+๐‘›)ฮฉ๎€ท1+๐œ†๐‘›๐‘ฮฉโˆ’1๎€ธ๎ƒชโˆ’โˆ‘(๐‘›+๐‘โˆ’๐›ผ)๐‘š๐‘˜=๐‘›+1๐‘Ž๐‘+๐‘˜๎ƒญโˆ’๐‘‘๐‘+๐‘š+1โˆ‘โˆž๐‘˜=๐‘š+1๐‘Ž๐‘+๐‘˜โ‰ค1(๐‘งโˆˆ๐‘ˆ;๐‘šโ‰ฅ๐‘›+1),(2.38) which shows that Reฮจ1(๐‘ง)>0. Thus, we obtain Reฮจ1๎‚ป๐‘‘(๐‘ง)=Re๐‘+๐‘š+1๎‚ธ๐‘“(๐‘ง)๐‘ ๐‘šโˆ’๎‚ต1(๐‘ง)1โˆ’๐‘‘๐‘+๐‘š+1๎‚ป๎‚ถ๎‚น๎‚ผ>0โŸนRe๐‘“(๐‘ง)๐‘ ๐‘š๎‚ผ1(๐‘ง)>1โˆ’๐‘‘๐‘+๐‘š+1(2.39) or ๎‚ปRe๐‘“(๐‘ง)๐‘ ๐‘š๎‚ผ๐‘(๐‘ง)>1โˆ’ฮฉ(๐‘โˆ’๐›ผ)(1โˆ’๐œ€)(๐‘+๐‘š+1)ฮฉ๎€บ1+๐œ†(๐‘š+1)๐‘ฮฉโˆ’1๎€ป(๐‘+๐‘š+1โˆ’๐›ผ).(2.40) Let ๐‘“(๐‘ง)=๐‘ง๐‘โˆ’๐‘ฮฉ(๐‘โˆ’๐›ผ)๐œ€(๐‘+๐‘›)ฮฉ๎€ท1+๐œ†๐‘›๐‘ฮฉโˆ’1๎€ธ๐‘ง(๐‘›+๐‘โˆ’๐›ผ)๐‘+๐‘›โˆ’1โˆ’๐‘ฮฉ(๐‘โˆ’๐›ผ)๐œ€/(๐‘+๐‘›)ฮฉ๎€ท1+๐œ†๐‘›๐‘ฮฉโˆ’1๎€ธ(๐‘›+๐‘โˆ’๐›ผ)๐‘‘๐‘+๐‘š+1๐‘ง๐‘+๐‘š+1.(2.41) Then, ๐‘“(๐‘ง) satisfies the condition (2.31) and ๐‘“โˆˆโ„‘โˆ—๐œ€(ฮฉ,๐œ†,๐‘,๐›ผ). Thus, we can write ๐‘“(๐‘ง)๐‘ ๐‘š=๎€ท๐‘(๐‘ง)1โˆ’ฮฉ(๐‘โˆ’๐›ผ)๐œ€/(๐‘+๐‘›)ฮฉ๎€ท1+๐œ†๐‘›๐‘ฮฉโˆ’1๎€ธ๎€ธ๐‘ง(๐‘›+๐‘โˆ’๐›ผ)๐‘›๎€ท๐‘1โˆ’ฮฉ(๐‘โˆ’๐›ผ)๐œ€/(๐‘+๐‘›)ฮฉ๎€ท1+๐œ†๐‘›๐‘ฮฉโˆ’1๎€ธ๎€ธ๐‘ง(๐‘›+๐‘โˆ’๐›ผ)๐‘›โˆ’๎€ท๎€ท๐‘1โˆ’ฮฉ(๐‘โˆ’๐›ผ)๐œ€/(๐‘+๐‘›)ฮฉ๎€ท1+๐œ†๐‘›๐‘ฮฉโˆ’1๎€ธ๎€ธ(๐‘›+๐‘โˆ’๐›ผ)/๐‘‘๐‘+๐‘š+1๎€ธ๐‘ง๐‘š+1๎€ท๐‘1โˆ’ฮฉ(๐‘โˆ’๐›ผ)๐œ€/(๐‘+๐‘›)ฮฉ๎€ท1+๐œ†๐‘›๐‘ฮฉโˆ’1๎€ธ๎€ธ๐‘ง(๐‘›+๐‘โˆ’๐›ผ)๐‘›๐‘=1โˆ’1โˆ’๎€ท๎€ทฮฉ(๐‘โˆ’๐›ผ)๐œ€/(๐‘+๐‘›)ฮฉ๎€ท1+๐œ†๐‘›๐‘ฮฉโˆ’1๎€ธ๎€ธ(๐‘›+๐‘โˆ’๐›ผ)/๐‘‘๐‘+๐‘š+1๎€ธ๐‘ง๐‘š+1๎€ท๐‘1โˆ’ฮฉ(๐‘โˆ’๐›ผ)๐œ€/(๐‘+๐‘›)ฮฉ๎€ท1+๐œ†๐‘›๐‘ฮฉโˆ’1๎€ธ๎€ธ๐‘ง(๐‘›+๐‘โˆ’๐›ผ)๐‘›(2.42) and taking as ๐‘งโ†’1โˆ’1โŸถ1โˆ’๐‘‘๐‘+๐‘š+1๐‘=1โˆ’ฮฉ(๐‘โˆ’๐›ผ)(1โˆ’๐œ€)(๐‘+๐‘š+1)ฮฉ๎€บ1+๐œ†(๐‘š+1)๐‘ฮฉโˆ’1๎€ป(๐‘+๐‘š+1โˆ’๐›ผ),(2.43) which shows that the bound in (2.32) is the best possible.
By using definitions of ๐‘“(๐‘ง) and ๐‘ ๐‘š(๐‘ง), we can write ๐‘ ๐‘š(๐‘ง)=๐‘ง๐‘“(๐‘ง)๐‘โˆ’๎€ท(๐‘โˆ’๐›ผ)๐œ€/((๐‘+๐‘›)/๐‘)ฮฉ๎€ท1+๐œ†๐‘›๐‘ฮฉโˆ’1๎€ธ๎€ธ๐‘ง(๐‘›+๐‘โˆ’๐›ผ)๐‘+๐‘›โˆ’โˆ‘๐‘š๐‘˜=๐‘›+1๐‘Ž๐‘+๐‘˜๐‘ง๐‘+๐‘˜๐‘ง๐‘โˆ’๎€ท(๐‘โˆ’๐›ผ)๐œ€/((๐‘+๐‘›)/๐‘)ฮฉ๎€ท1+๐œ†๐‘›๐‘ฮฉโˆ’1๎€ธ๎€ธ๐‘ง(๐‘›+๐‘โˆ’๐›ผ)๐‘+๐‘›โˆ’โˆ‘โˆž๐‘˜=๐‘›+1๐‘Ž๐‘+๐‘˜๐‘ง๐‘+๐‘˜=๐‘ง๐‘โˆ’๎€ท(๐‘โˆ’๐›ผ)๐œ€/((๐‘+๐‘›)/๐‘)ฮฉ๎€ท1+๐œ†๐‘›๐‘ฮฉโˆ’1๎€ธ๎€ธ๐‘ง(๐‘›+๐‘โˆ’๐›ผ)๐‘+๐‘›๐‘ง๐‘โˆ’๎€ท(๐‘โˆ’๐›ผ)๐œ€/((๐‘+๐‘›)/๐‘)ฮฉ๎€ท1+๐œ†๐‘›๐‘ฮฉโˆ’1๎€ธ๎€ธ๐‘ง(๐‘›+๐‘โˆ’๐›ผ)๐‘+๐‘›โˆ’โˆ‘โˆž๐‘˜=๐‘›+1๐‘Ž๐‘+๐‘˜๐‘ง๐‘+๐‘˜โˆ’โˆ‘โˆž๐‘˜=๐‘›+1๐‘Ž๐‘+๐‘˜๐‘ง๐‘+๐‘˜+โˆ‘โˆž๐‘˜=๐‘š+1๐‘Ž๐‘+๐‘˜๐‘ง๐‘+๐‘˜๐‘ง๐‘โˆ’๎€ท(๐‘โˆ’๐›ผ)๐œ€/((๐‘+๐‘›)/๐‘)ฮฉ๎€ท1+๐œ†๐‘›๐‘ฮฉโˆ’1๎€ธ๎€ธ๐‘ง(๐‘›+๐‘โˆ’๐›ผ)๐‘+๐‘›โˆ’โˆ‘โˆž๐‘˜=๐‘›+1๐‘Ž๐‘+๐‘˜๐‘ง๐‘+๐‘˜โˆ‘=1+โˆž๐‘˜=๐‘š+1๐‘Ž๐‘+๐‘˜๐‘ง๐‘˜๎€ท1โˆ’(๐‘โˆ’๐›ผ)๐œ€/((๐‘+๐‘›)/๐‘)ฮฉ๎€ท1+๐œ†๐‘›๐‘ฮฉโˆ’1๎€ธ๎€ธ๐‘ง(๐‘›+๐‘โˆ’๐›ผ)๐‘›โˆ’โˆ‘โˆž๐‘˜=๐‘›+1๐‘Ž๐‘+๐‘˜๐‘ง๐‘˜.(2.44) If we put ๐œ“2๎€ท(๐‘ง)=1+๐‘‘๐‘+๐‘š+1๎€ธ๎‚ป๐‘ ๐‘š(๐‘ง)๐‘“โˆ’๐‘‘(๐‘ง)๐‘+๐‘š+11+๐‘‘๐‘+๐‘š+1๎‚ผ๎ƒฉ๎€ท=1+1+๐‘‘๐‘+๐‘š+1๎€ธโˆ‘โˆž๐‘˜=๐‘š+1๐‘Ž๐‘+๐‘˜๐‘ง๐‘˜๎€ท1โˆ’(๐‘โˆ’๐›ผ)๐œ€/((๐‘+๐‘›)/๐‘)ฮฉ๎€ท1+๐œ†๐‘›๐‘ฮฉโˆ’1๎€ธ๎€ธ๐‘ง(๐‘›+๐‘โˆ’๐›ผ)๐‘›โˆ’โˆ‘โˆž๐‘˜=๐‘›+1๐‘Ž๐‘+๐‘˜๐‘ง๐‘˜๎ƒช(2.45) and make use of (2.35), we can deduce that ||||๐œ“2(๐‘ง)โˆ’1๐œ“2||||=|||||||||||||||||๎€ท(๐‘ง)+11+๐‘‘๐‘+๐‘š+1๎€ธโˆ‘โˆž๐‘˜=๐‘š+1๐‘Ž๐‘+๐‘˜๐‘ง๐‘˜โŽ›โŽœโŽœโŽœโŽœโŽ2โˆ’2(๐‘โˆ’๐›ผ)๐œ€๎‚ต๐‘+๐‘›๐‘๎‚ถฮฉ๎€ท1+๐œ†๐‘›๐‘ฮฉโˆ’1๎€ธโŽžโŽŸโŽŸโŽŸโŽŸโŽ ๐‘ง(๐‘›+๐‘โˆ’๐›ผ)๐‘›โˆ‘โˆ’2๐‘š๐‘˜=๐‘›+1๐‘Ž๐‘+๐‘˜๐‘ง๐‘˜+๎€ท๐‘‘๐‘+๐‘š+1๎€ธโˆ‘โˆ’1โˆž๐‘˜=๐‘š+1๐‘Ž๐‘+๐‘˜๐‘ง๐‘˜|||||||||||||||||โ‰ค๎€ท1+๐‘‘๐‘+๐‘š+1๎€ธโˆ‘โˆž๐‘˜=๐‘š+1๐‘Ž๐‘+๐‘˜โŽ›โŽœโŽœโŽœโŽœโŽ2โˆ’2(๐‘โˆ’๐›ผ)๐œ€๎‚ต๐‘+๐‘›๐‘๎‚ถฮฉ๎€ท1+๐œ†๐‘›๐‘ฮฉโˆ’1๎€ธโŽžโŽŸโŽŸโŽŸโŽŸโŽ โˆ‘(๐‘›+๐‘โˆ’๐›ผ)โˆ’2๐‘š๐‘˜=๐‘›+1๐‘Ž๐‘+๐‘˜โˆ’๎€ท๐‘‘๐‘+๐‘š+1๎€ธโˆ‘โˆ’1โˆž๐‘˜=๐‘š+1๐‘Ž๐‘+๐‘˜=๎€ท1+๐‘‘๐‘+๐‘š+1๎€ธโˆ‘โˆž๐‘˜=๐‘š+1๐‘Ž๐‘+๐‘˜2โŽงโŽชโŽชโŽจโŽชโŽชโŽฉโŽ›โŽœโŽœโŽœโŽœโŽ1โˆ’(๐‘โˆ’๐›ผ)๐œ€๎‚ต๐‘+๐‘›๐‘๎‚ถฮฉ๎€ท1+๐œ†๐‘›๐‘ฮฉโˆ’1๎€ธโŽžโŽŸโŽŸโŽŸโŽŸโŽ โˆ’โˆ‘(๐‘›+๐‘โˆ’๐›ผ)๐‘š๐‘˜=๐‘›+1๐‘Ž๐‘+๐‘˜โŽซโŽชโŽชโŽฌโŽชโŽชโŽญโˆ’๎€ท๐‘‘๐‘+๐‘š+1๎€ธโˆ‘โˆ’1โˆž๐‘˜=๐‘š+1๐‘Ž๐‘+๐‘˜โ‰ค๎€ท1+๐‘‘๐‘+๐‘š+1๎€ธโˆ‘โˆž๐‘˜=๐‘š+1๐‘Ž๐‘+๐‘˜2๐‘‘๐‘+๐‘š+1โˆ‘โˆž๐‘˜=๐‘š+1๐‘Ž๐‘+๐‘˜โˆ’๎€ท๐‘‘๐‘+๐‘š+1๎€ธโˆ‘โˆ’1โˆžk=๐‘š+1๐‘Ž๐‘+๐‘˜=1,(2.46)|(๐œ“2(๐‘ง)โˆ’1)/(๐œ“2(๐‘ง)+1)|โ‰ค1 requires that Re{๐œ“2(๐‘ง)}>0. Thus, we obtain ๎€ฝฮจRe2๎€พ=๎€ท(๐‘ง)1+๐‘‘๐‘+๐‘š+1๎€ธ๎‚ธ๐‘ Re๐‘š(๐‘ง)๐‘“โˆ’๎‚ต๐‘‘(๐‘ง)๐‘+๐‘š+11+๐‘‘๐‘+๐‘š+1๎‚ถ๎‚น>0(2.47) or ๎‚ป๐‘ Re๐‘š(๐‘ง)๐‘“๎‚ผ>๎‚ต๐‘‘(๐‘ง)๐‘+๐‘š+11+๐‘‘๐‘+๐‘š+1๎‚ถ.(2.48) It follows from the last inequality that assertion (2.33) of the Theorem 2.5 holds.
The bound in (2.33) is sharp with the extremal function given by (2.41). The proof of the theorem is thus completed.

If ฮฉ=0, ๐œ†=0, and ๐‘›=1 are taken in Theorem 2.5, the following result is obtained given by Liu [6].

Corollary 2.6. Let ๐‘“ be given by (1.19) and define the partial sums ๐‘ ๐‘š(๐‘ง) by ๐‘ ๐‘šโŽงโŽชโŽชโŽจโŽชโŽชโŽฉ๐‘ง(๐‘ง)=๐‘โˆ’(๐‘โˆ’๐›ผ)๐œ€๐‘ง๐‘+1โˆ’๐›ผ๐‘+1๐‘ง,๐‘š=1,๐‘โˆ’(๐‘โˆ’๐›ผ)๐œ€๐‘ง(๐‘›+๐‘โˆ’๐›ผ)๐‘+1โˆ’๐‘š๎“๐‘˜=2๐‘Ž๐‘+๐‘˜๐‘ง๐‘+๐‘˜,๐‘š=2,3,โ€ฆ.(2.49) Suppose also that โˆž๎“๐‘˜=2๐‘‘๐‘+๐‘˜๐‘Ž๐‘+๐‘˜โ‰ค1โˆ’(๐‘โˆ’๐›ผ)๐œ€๎‚ต๐‘+1โˆ’๐›ผwhere๐‘‘๐‘+๐‘˜=๐‘˜+๐‘โˆ’๐›ผ๎‚ถ(๐‘โˆ’๐›ผ)(1โˆ’๐œ€);0โ‰ค๐›ผ<๐‘,0โ‰ค๐œ€<1.(2.50) Then, for ๐‘šโ‰ฅ2, one has ๎‚ปRe๐‘“(๐‘ง)๐‘ ๐‘š๎‚ผ>(๐‘ง)(๐‘โˆ’๐›ผ)๐œ€+๐‘š+1๎‚ต๐‘ ๐‘+๐‘š+1โˆ’๐›ผRe๐‘š(๐‘ง)๎‚ถ>๐‘“(๐‘ง)๐‘+๐‘š+1โˆ’๐›ผ(.๐‘โˆ’๐›ผ)(2โˆ’๐œ€)+๐‘š+1(2.51) Each of the bounds in (2.51) is the best possible.

References

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