Abstract

The main purpose of this paper is to derive some coefficient inequalities and subordination properties for certain subclasses of analytic functions involving the Salagean operator. Relevant connections of the results presented here with those obtained in earlier works are also pointed out.

1. Introduction

Let ๐’œ denote the class of functions of the form๐‘“(๐‘ง)=๐‘ง+โˆž๎“๐‘—=2๐‘Ž๐‘—๐‘ง๐‘—,(1.1) which are analytic in the open unit disk ๐•Œโˆถ={๐‘งโˆถ๐‘งโˆˆโ„‚,|๐‘ง|<1}.(1.2)

For 0โ‰ฆ๐›ผ<1, we denote by ๐’ฎโˆ—(๐›ผ) and ๐’ฆ(๐›ผ) the usual subclasses of ๐’œ consisting of functions which are, respectively, starlike of order ๐›ผ and convex of order ๐›ผ in ๐•Œ. Clearly, we know that ๐‘“โˆˆ๐’ฆ(๐›ผ)โŸบ๐‘ง๐‘“โ€ฒโˆˆ๐’ฎโˆ—(๐›ผ).(1.3)

A function ๐‘“โˆˆ๐’œ is said to be in the class โ„ณ(๐›ฝ) if it satisfies the inequality โ„œ๎‚ต๐‘ง๐‘“๎…ž(๐‘ง)๎‚ถ๐‘“(๐‘ง)<๐›ฝ(๐‘งโˆˆ๐•Œ),(1.4) for some ๐›ฝ(๐›ฝ>1). Also, a function ๐‘“โˆˆ๐’œ is said to be in the class ๐’ฉ(๐›ฝ) if and only if ๐‘ง๐‘“โ€ฒโˆˆโ„ณ(๐›ฝ). The classes โ„ณ(๐›ฝ) and ๐’ฉ(๐›ฝ) were introduced and investigated recently by Owa and Srivastava [1] (see also Nishiwaki and Owa [2], Owa and Nishiwaki [3], and Srivastava and Attiya [4]).

Sฤƒlฤƒgean [5] introduced the operator ๐ท0๐‘“(๐‘ง)=๐‘“(๐‘ง),๐ท1๐‘“(๐‘ง)=๐ท๐‘“(๐‘ง)=๐‘ง๐‘“๎…ž๐ท(๐‘ง),๐‘›๎€ท๐ท๐‘“(๐‘ง)=๐ท๐‘›โˆ’1๎€ธ๐‘“(๐‘ง)(๐‘›โˆˆโ„•โˆถ={1,2,โ€ฆ}).(1.5) We note that ๐ท๐‘›๐‘“(๐‘ง)=๐‘ง+โˆž๎“๐‘—=2๐‘—๐‘›๐‘Ž๐‘—๐‘ง๐‘—๎€ท๐‘›โˆˆโ„•0๎€ธ.โˆถ=โ„•โˆช{0}(1.6)

Given two functions ๐‘“,๐‘”โˆˆ๐ด, where ๐‘“ is given by (1.1) and ๐‘” is defined by ๐‘”(๐‘ง)=๐‘ง+โˆž๎“๐‘›=2๐‘๐‘›๐‘ง๐‘›,(1.7) the Hadamard product (or convolution) ๐‘“โˆ—๐‘” is defined by (๐‘“โˆ—๐‘”)(๐‘ง)โˆถ=๐‘ง+โˆž๎“๐‘›=2๐‘Ž๐‘›๐‘๐‘›๐‘ง๐‘›=โˆถ(๐‘”โˆ—๐‘“)(๐‘ง).(1.8)

For two functions ๐‘“ and ๐‘”, analytic in ๐•Œ, we say that the function ๐‘“ is subordinate to ๐‘” in ๐•Œ, and write ๐‘“(๐‘ง)โ‰บ๐‘”(๐‘ง)(๐‘งโˆˆ๐•Œ)(1.9) if there exists a Schwarz function ๐œ”, which is analytic in ๐•Œ with ||||๐œ”(0)=0,๐œ”(๐‘ง)<1(๐‘งโˆˆ๐•Œ)(1.10) such that ๐‘“(๐‘ง)=๐‘”(๐œ”(๐‘ง))(๐‘งโˆˆ๐•Œ).(1.11) Indeed, it is known that ๐‘“(๐‘ง)โ‰บ๐‘”(๐‘ง),(๐‘งโˆˆ๐•Œ)โŸน๐‘“(0)=๐‘”(0),๐‘“(๐•Œ)โŠ‚๐‘”(๐•Œ).(1.12) Furthermore, if the function ๐‘” is univalent in ๐•Œ, then we have the following equivalence: ๐‘“(๐‘ง)โ‰บ๐‘”(๐‘ง),(๐‘งโˆˆ๐•Œ)โŸบ๐‘“(0)=๐‘”(0),๐‘“(๐•Œ)โŠ‚๐‘”(๐•Œ).(1.13)

In recent years, Deng [6] (see also Kamali [7], AltintaลŸ et al. [8], Srivastava et al. [9], and Xu et al. [10]) introduced and investigated the following subclass of ๐’œ involving the S Sฤƒlฤƒgean lagean operator and obtained the coefficient bounds for this function class.

Definition 1.1. A function ๐‘“โˆˆ๐’œ is said to be in the class ๐’ฎ๐‘›(๐œ†,๐›ผ,๐‘) if it satisfies the inequality โ„œ๎‚ต11+๐‘๎‚ต(1โˆ’๐œ†)๐ท๐‘›+1๐‘“(๐‘ง)+๐œ†๐ท๐‘›+2๐‘“(๐‘ง)(1โˆ’๐œ†)๐ท๐‘›๐‘“(๐‘ง)+๐œ†๐ท๐‘›+1๐‘“(๐‘ง)โˆ’1๎‚ถ๎‚ถ>๐›ผ(๐‘งโˆˆ๐•Œ),(1.14) where ๐‘›โˆˆโ„•0,๐‘โˆˆโ„‚โงต{0},0โ‰ฆ๐›ผ<1,0โ‰ฆ๐œ†โ‰ฆ1.(1.15)
It is easy to see that the class ๐’ฎ๐‘›(๐œ†,๐›ผ,๐‘) includes the classes ๐’ฎโˆ—(๐›ผ) and ๐’ฆ(๐›ผ) as its special cases.
Now, motivated essentially by the above-mentioned function classes, we introduce the following subclass of ๐’œ of analytic functions.

Definition 1.2. A function ๐‘“โˆˆ๐’œ is said to be in the class โ„ณ๐‘›(๐œ†,๐›ฝ,๐‘) if it satisfies the inequality: โ„œ๎‚ต11+๐‘๎‚ต(1โˆ’๐œ†)๐ท๐‘›+1๐‘“(๐‘ง)+๐œ†๐ท๐‘›+2๐‘“(๐‘ง)(1โˆ’๐œ†)๐ท๐‘›๐‘“(๐‘ง)+๐œ†๐ท๐‘›+1๐‘“(๐‘ง)โˆ’1๎‚ถ๎‚ถ<๐›ฝ(๐‘งโˆˆ๐•Œ),(1.16) where ๐‘›โˆˆโ„•0,๐‘โˆˆโ„‚โงต{0},๐›ฝ>1,0โ‰ฆ๐œ†โ‰ฆ1.(1.17)

It is also easy to see that the classes โ„ณ(๐›ฝ) and ๐’ฉ(๐›ฝ) are special cases of the class โ„ณ๐‘›(๐œ†,๐›ฝ,๐‘).

In this paper, we aim at proving some coefficient inequalities and subordination properties for the classes ๐’ฎ๐‘›(๐œ†,๐›ฝ,๐‘) and โ„ณ๐‘›(๐œ†,๐›ฝ,๐‘). The results presented here would provide extensions of those given in earlier works. Several other new results are also obtained.

2. Coefficient Inequalities

In this section, we derive some coefficient inequalities for the classes ๐’ฎ๐‘›(๐œ†,๐›ผ,๐‘) and โ„ณ๐‘›(๐œ†,๐›ผ,๐‘).

Theorem 2.1. Let ๐‘›โˆˆโ„•0,๐‘โˆˆโ„‚โงต{0},0โ‰ฆ๐›ผ<1,0โ‰ฆ๐œ†โ‰ฆ1.(2.1) If ๐‘“โˆˆ๐’œ satisfies the coefficient inequality โˆž๎“๐‘—=2๎€บ(1โˆ’๐œ†)๐‘—๐‘›+๐œ†๐‘—๐‘›+1||๐‘||(๎€ป||๐‘Ž๎€ป๎€บ๐‘—โˆ’1+1โˆ’๐›ผ)๐‘—||โ‰ฆ||๐‘||(1โˆ’๐›ผ),(2.2) then ๐‘“โˆˆ๐’ฎ๐‘›(๐œ†,๐›ผ,๐‘).

Proof. To prove ๐‘“โˆˆ๐’ฎ๐‘›(๐œ†,๐›ผ,๐‘), it is sufficient to show that ||||(1โˆ’๐œ†)๐ท๐‘›+1๐‘“(๐‘ง)+๐œ†๐ท๐‘›+2๐‘“(๐‘ง)(1โˆ’๐œ†)๐ท๐‘›๐‘“(๐‘ง)+๐œ†๐ท๐‘›+1๐‘“||||<||๐‘||(๐‘ง)โˆ’1(1โˆ’๐›ผ)(๐‘งโˆˆ๐•Œ).(2.3) By noting that ||||(1โˆ’๐œ†)๐ท๐‘›+1๐‘“(๐‘ง)+๐œ†๐ท๐‘›+2๐‘“(๐‘ง)(1โˆ’๐œ†)๐ท๐‘›๐‘“(๐‘ง)+๐œ†๐ท๐‘›+1๐‘“||||=|||||โˆ‘(๐‘ง)โˆ’1โˆž๐‘—=2๎€บ(๎€ท๐‘—1โˆ’๐œ†)๐‘›+1โˆ’๐‘—๐‘›๎€ธ๎€ท๐‘—+๐œ†๐‘›+2โˆ’๐‘—๐‘›+1๐‘Ž๎€ธ๎€ป๐‘—๐‘ง๐‘—โˆ’1โˆ‘1+โˆž๐‘—=2๎€บ(1โˆ’๐œ†)๐‘—๐‘›+๐œ†๐‘—๐‘›+1๎€ป๐‘Ž๐‘—๐‘ง๐‘—โˆ’1|||||โ‰ฆโˆ‘โˆž๐‘—=2๎€บ๎€ท๐‘—(1โˆ’๐œ†)๐‘›+1โˆ’๐‘—๐‘›๎€ธ๎€ท๐‘—+๐œ†๐‘›+2โˆ’๐‘—๐‘›+1||๐‘Ž๎€ธ๎€ป๐‘—||โˆ‘1โˆ’โˆž๐‘—=2๎€บ(1โˆ’๐œ†)๐‘—๐‘›+๐œ†๐‘—๐‘›+1๎€ป||๐‘Ž๐‘—||,(2.4) it follows from (2.2) that the above last expression is bounded by |๐‘|(1โˆ’๐›ผ). This completes the proof of Theorem 2.1.

Theorem 2.2. Let ๐‘›โˆˆโ„•0,๐‘โˆˆโ„‚โงต{0},๐›ฝ>1,0โ‰ฆ๐œ†โ‰ฆ1.(2.5) If ๐‘“โˆˆ๐’œ satisfies the coefficient inequality โˆž๎“๐‘—=2๎€บ(1โˆ’๐œ†)๐‘—๐‘›+๐œ†๐‘—๐‘›+1||||||||๎€ธ||๐‘Ž๎€ป๎€ท๐‘โˆ’1+๐‘—+๐‘—โˆ’1โˆ’(2๐›ฝโˆ’1)๐‘๐‘—||||๐‘||(โ‰ฆ2๐›ฝโˆ’1),(2.6) then ๐‘“โˆˆโ„ณ๐‘›(๐œ†,๐›ฝ).

Proof. To prove ๐‘“โˆˆโ„ณ๐‘›(๐œ†,๐›ฝ,๐‘), it suffices to show that ||||11+๐‘๎‚ต(1โˆ’๐œ†)๐ท๐‘›+1๐‘“(๐‘ง)+๐œ†๐ท๐‘›+2๐‘“(๐‘ง)(1โˆ’๐œ†)๐ท๐‘›๐‘“(๐‘ง)+๐œ†๐ท๐‘›+1๐‘“๎‚ถ||||<||||1(๐‘ง)โˆ’11+๐‘๎‚ต(1โˆ’๐œ†)๐ท๐‘›+1๐‘“(๐‘ง)+๐œ†๐ท๐‘›+2๐‘“(๐‘ง)(1โˆ’๐œ†)๐ท๐‘›๐‘“(๐‘ง)+๐œ†๐ท๐‘›+1๎‚ถ||||.๐‘“(๐‘ง)โˆ’1โˆ’2๐›ฝ(2.7) We consider ๐‘€โˆˆโ„ defined by ||๎€บ๐‘€โˆถ=(๐‘โˆ’1)(1โˆ’๐œ†)๐ท๐‘›๐‘“(๐‘ง)+๐œ†๐ท๐‘›+1๎€ป๐‘“(๐‘ง)+(1โˆ’๐œ†)๐ท๐‘›+1๐‘“(๐‘ง)+๐œ†๐ท๐‘›+2||โˆ’||๐‘“(๐‘ง)(1โˆ’๐œ†)๐ท๐‘›+1๐‘“(๐‘ง)+๐œ†๐ท๐‘›+2๐‘“[]๎€บ(๐‘ง)โˆ’(2๐›ฝโˆ’1)๐‘+1(1โˆ’๐œ†)๐ท๐‘›๐‘“(๐‘ง)+๐œ†๐ท๐‘›+1๐‘“๎€ป||=|||||(๐‘ง)๐‘๐‘ง+โˆž๎“๐‘—=2๎€ฝ๎€บ(๐‘โˆ’1)(1โˆ’๐œ†)๐‘—๐‘›+๐œ†๐‘—๐‘›+1๎€ป+๎€บ(1โˆ’๐œ†)๐‘—๐‘›+1+๐œ†๐‘—๐‘›+2๐‘Ž๎€ป๎€พ๐‘—๐‘ง๐‘—|||||โˆ’|||||๐‘ง+โˆž๎“๐‘—=2๎€บ(1โˆ’๐œ†)๐‘—๐‘›+1+๐œ†๐‘—๐‘›+2๎€ป๐‘Ž๐‘—๐‘ง๐‘—โˆ’[]๎ƒฉ(2๐›ฝโˆ’1)๐‘+1๐‘ง+โˆž๎“๐‘—=2๎€บ(1โˆ’๐œ†)๐‘—๐‘›+๐œ†๐‘—๐‘›+1๎€ป๐‘Ž๐‘—๐‘ง๐‘—๎ƒช|||||.(2.8) Thus, for |๐‘ง|=๐‘Ÿ<1, we have ||๐‘||๐‘€โ‰ฆ๐‘Ÿ+โˆž๎“๐‘—=2๎€ฝ||||๎€บ(๐‘โˆ’11โˆ’๐œ†)๐‘—๐‘›+๐œ†๐‘—๐‘›+1๎€ป+(1โˆ’๐œ†)๐‘—๐‘›+1+๐œ†๐‘—๐‘›+2๎€พ||๐‘Ž๐‘—||๐‘Ÿ๐‘—โˆ’๎ƒฌ||๐‘||(2๐›ฝโˆ’1)๐‘Ÿโˆ’โˆž๎“๐‘—=2||๎€บ(1โˆ’๐œ†)๐‘—๐‘›+1+๐œ†๐‘—๐‘›+2๎€ปโˆ’[]๎€บ(2๐›ฝโˆ’1)๐‘+1(1โˆ’๐œ†)๐‘—๐‘›+๐œ†๐‘—๐‘›+1๎€ป||||๐‘Ž๐‘—||๐‘Ÿ๐‘—๎ƒญ<๎ƒฉโˆž๎“๐‘—=2๎€ฝ||||๎€บ๐‘โˆ’1(1โˆ’๐œ†)๐‘—๐‘›+๐œ†๐‘—๐‘›+1๎€ป+(1โˆ’๐œ†)๐‘—๐‘›+1+๐œ†๐‘—๐‘›+2+||๎€บ(1โˆ’๐œ†)๐‘—๐‘›+1+๐œ†๐‘—๐‘›+2๎€ปโˆ’[]๎€บ(2๐›ฝโˆ’1)๐‘+1(1โˆ’๐œ†)๐‘—๐‘›+๐œ†๐‘—๐‘›+1๎€ป||๎€พ||๐‘Ž๐‘—||||๐‘||๎ƒชโˆ’2(๐›ฝโˆ’1)๐‘Ÿ.(2.9) It follows from (2.6) that ๐‘€<0, which implies that (2.7) holds, that is, ๐‘“โˆˆโ„ณ๐‘›(๐œ†,๐›ฝ,๐‘). The proof of Theorem 2.2 is evidently completed.

To prove our next result, we need the following lemma.

Lemma 2.3. Let ๐›ฝ>1 and ๐‘โˆˆโ„‚โงต{0}. Suppose also that the sequence {๐”น๐‘—}โˆž๐‘—=1 is defined by ๐”น1๐”น=1(๐‘—=1),๐‘—=2||๐‘||(๐›ฝโˆ’1)๐‘—โˆ’1๐‘—โˆ’1๎“๐‘˜=1๐”น๐‘˜(๐‘—โˆˆโ„•โงต{1}),(2.10) then ๐”น๐‘—=1(๐‘—โˆ’1)!๐‘—โˆ’2๎‘๐‘˜=0๎€บ2||๐‘||๎€ป(๐›ฝโˆ’1)+๐‘˜(๐‘—โˆˆโ„•โงต{1}).(2.11)

Proof. We make use of the principle of mathematical induction to prove the assertion (2.11) of Lemma 2.3. Indeed, from (2.10), we know that ๐”น2||๐‘||1=2(๐›ฝโˆ’1)=1!0๎‘๐‘˜=0๎€บ2||๐‘||๎€ป,(๐›ฝโˆ’1)+๐‘˜(2.12) which implies that (2.11) holds for ๐‘—=2.
We now suppose that (2.11) holds for ๐‘—=๐‘š(๐‘šโ‰ง2), then ๐”น๐‘š=1(๐‘šโˆ’1)!๐‘šโˆ’2๎‘๐‘˜=0๎€บ2||๐‘||๎€ป.(๐›ฝโˆ’1)+๐‘˜(2.13) Combining (2.10) and (2.13), we find that ๐”น๐‘š+1=2||๐‘||(๐›ฝโˆ’1)๐‘š๐‘š๎“๐‘˜=1๐”น๐‘˜=2||๐‘||(๐›ฝโˆ’1)๐‘š๐‘šโˆ’1๎“๐‘˜=1๐”น๐‘˜+2||๐‘||(๐›ฝโˆ’1)๐‘š๐”น๐‘š=2||๐‘||(๐›ฝโˆ’1)๐‘šโ‹…๐‘šโˆ’12||๐‘||๐”น(๐›ฝโˆ’1)๐‘š+2||๐‘||(๐›ฝโˆ’1)๐‘š๐”น๐‘š=2||๐‘||(๐›ฝโˆ’1)+๐‘šโˆ’1๐‘š๐”น๐‘š=1๐‘š!๐‘šโˆ’1๎‘๐‘˜=0๎€บ2||๐‘||๎€ป,(๐›ฝโˆ’1)+๐‘˜(2.14) which shows that (2.11) holds for ๐‘—=๐‘š+1. The proof of Lemma 2.3 is evidently completed.

Theorem 2.4. Let ๐‘“โˆˆโ„ณ๐‘›(๐œ†,๐›ฝ,๐‘), then ||๐‘Ž๐‘—||โ‰ฆ1(๐‘—โˆ’1)!(1โˆ’๐œ†+๐œ†๐‘—)๐‘—๐‘›๐‘—โˆ’2๎‘๐‘˜=0๎€บ2||๐‘||๎€ป(๐›ฝโˆ’1)+๐‘˜(๐‘—โˆˆโ„•โงต{1}).(2.15)

Proof. We first suppose that ๐น(๐‘ง)โˆถ=(1โˆ’๐œ†)๐ท๐‘›๐‘“(๐‘ง)+๐œ†๐ท๐‘›+1๐‘“(๐‘ง)=๐‘ง+โˆž๎“๐‘—=2๐ต๐‘—๐‘ง๐‘—(๐‘งโˆˆ๐•Œ;๐‘“โˆˆ๐’œ),(2.16) where ๐ต๐‘—=๐‘—๐‘›(1โˆ’๐œ†+๐œ†๐‘—)๐‘Ž๐‘—.(2.17) Next, by setting ๎€ทโ„Ž(๐‘ง)โˆถ=๐›ฝโˆ’1โˆ’(1/๐‘)๐‘ง๐น๎…ž๎€ธ(๐‘ง)/๐น(๐‘ง)โˆ’1๐›ฝโˆ’1=1+โ„Ž1๐‘ง+โ„Ž2๐‘ง2๎€ท+โ‹ฏ๐‘งโˆˆ๐•Œ;๐‘“โˆˆโ„ณ๐‘›๎€ธ,(๐œ†,๐›ฝ,๐‘)(2.18) we easily find that โ„Žโˆˆ๐’ซ. It follows from (2.18) that ๐‘ง๐น๎…ž[]๐น(๐‘ง)=1+๐‘(๐›ฝโˆ’1)(๐‘ง)โˆ’๐‘(๐›ฝโˆ’1)โ„Ž(๐‘ง)๐น(๐‘ง).(2.19) We now find from (2.16), (2.18), and (2.19) that ๐‘ง+2๐ต2๐‘ง2+โ‹ฏ+๐‘—๐ต๐‘—๐‘ง๐‘—=[]๎€ท+โ‹ฏ1+๐‘(๐›ฝโˆ’1)๐‘ง+๐ต2๐‘ง2+โ‹ฏ+๐ต๐‘—๐‘ง๐‘—๎€ธ๎€ท+โ‹ฏโˆ’๐‘(๐›ฝโˆ’1)1+โ„Ž1๐‘ง+โ„Ž2๐‘ง2+โ‹ฏ+โ„Ž๐‘—๐‘ง๐‘—+โ‹ฏ๎€ธ๎€ท๐‘ง+๐ต2๐‘ง2+โ‹ฏ+๐ต๐‘—๐‘ง๐‘—๎€ธ.+โ‹ฏ(2.20) By evaluating the coefficients of ๐‘ง๐‘— in both the sides of (2.20), we get ๐‘—๐ต๐‘—=[]๐ต1+๐‘(๐›ฝโˆ’1)๐‘—๎€ทโ„Žโˆ’๐‘(๐›ฝโˆ’1)๐‘—โˆ’1+โ„Ž๐‘—โˆ’2๐ต2+โ‹ฏ+โ„Ž1๐ต๐‘—โˆ’1+๐ต๐‘—๎€ธ.(2.21) On the other hand, it is well known that ||โ„Ž๐‘˜||โ‰ฆ2(๐‘˜โˆˆโ„•).(2.22) Combining (2.21) and (2.22), we easily get ||๐ต๐‘—||โ‰ฆ2||๐‘||(๐›ฝโˆ’1)๐‘—โˆ’1๐‘—โˆ’1๎“๐‘˜=1||๐ต๐‘˜||๎€ท๐ต1๎€ธ.=1;๐‘—โˆˆโ„•โงต{1}(2.23)
Suppose that ๐›ฝ>1 and ๐‘โˆˆโ„‚โงต{0}. We define the sequence {๐”น๐‘—}โˆž๐‘—=1 as follows: ๐”น1๐”น=1(๐‘—=1),๐‘—=2||๐‘||(๐›ฝโˆ’1)๐‘—โˆ’1๐‘—โˆ’1๎“๐‘˜=1๐”น๐‘˜(๐‘—โˆˆโ„•โงต{1}).(2.24) In order to prove that ||๐ต๐‘—||โ‰ฆ๐”น๐‘—(๐‘—โˆˆโ„•โงต{1}),(2.25) we use the principle of mathematical induction. By noting that ||๐ต2||||๐‘||โ‰ฆ2(๐›ฝโˆ’1),(2.26) thus, assuming that ||๐ต๐‘š||โ‰ฆ๐”น๐‘š(๐‘šโˆˆ{2,3,โ€ฆ,๐‘—}),(2.27) we find from (2.23) and (2.24) that ||๐ต๐‘—+1||โ‰ฆ2||๐‘||(๐›ฝโˆ’1)๐‘—๐‘—๎“๐‘˜=1||๐ต๐‘˜||โ‰ฆ2||๐‘||(๐›ฝโˆ’1)๐‘—๐‘—๎“๐‘˜=1๐”น๐‘˜=๐”น๐‘—+1(๐‘—โˆˆโ„•).(2.28) Therefore, by the principle of mathematical induction, we have ||๐ต๐‘—||โ‰ฆ๐”น๐‘—(๐‘—โˆˆโ„•โงต{1})(2.29) as desired.
By virtue of Lemma 2.3 and (2.24), we know that ๐”น๐‘—=1(๐‘—โˆ’1)!๐‘—โˆ’2๎‘๐‘˜=0๎€บ2||๐‘||๎€ป(๐›ฝโˆ’1)+๐‘˜(๐‘—โˆˆโ„•โงต{1}).(2.30) Combining (2.17), (2.29), and (2.30), we readily arrive at the coefficient estimates (2.15) asserted by Theorem 2.4.

Remark 2.5. Setting ๐œ†=0, ๐‘=1, and ๐‘›=0or1 in Theorem 2.4, we get the corresponding results obtained by Owa and Nishiwaki [3].

Remark 2.6. We cannot show that the result of Theorem 2.4 is sharp. Indeed, if one can prove the sharpness of Theorem 2.4, the sharpness of the corresponding result obtained by Deng [6] follows easily.

3. Subordination Properties

In view of Theorems 2.1 and 2.2, we now introduce the following subclasses: ๎‚‹๐’ฎ๐‘›(๐œ†,๐›ผ,๐‘)โŠ‚๐’ฎ๐‘›๎‚‹โ„ณ(๐œ†,๐›ผ,๐‘),๐‘›(๐œ†,๐›ฝ,๐‘)โŠ‚โ„ณ๐‘›(๐œ†,๐›ฝ,๐‘),(3.1) which consist of functions ๐‘“โˆˆ๐’œ whose Taylor-Maclaurin coefficients satisfy the inequalities (2.2) and (2.6), respectively.

A sequence {๐‘๐‘—}โˆž๐‘—=1 of complex numbers is said to be a subordinating factor sequence if, whenever ๐‘“ of the form (1.1) is analytic, univalent, and convex in ๐•Œ, we have the subordination โˆž๎“๐‘—=1๐‘Ž๐‘—๐‘๐‘—๐‘ง๐‘—๎€ท๐‘Žโ‰บ๐‘“(๐‘ง)1๎€ธ.=1;๐‘งโˆˆ๐•Œ(3.2)

To derive the subordination properties for the classes ๎‚‹๐’ฎ๐‘›(๐œ†,๐›ผ,๐‘) and ๎‚‹โ„ณ๐‘›(๐œ†,๐›ผ,๐‘), we need the following lemma.

Lemma 3.1 (see [11]). The sequence {๐‘๐‘—}โˆž๐‘—=1 is a subordinating factor sequence if and only if โ„œ๎ƒฉ1+2โˆž๎“๐‘—=1๐‘๐‘—๐‘ง๐‘—๎ƒช>0(๐‘งโˆˆ๐•Œ).(3.3)

Theorem 3.2. If ๎‚‹๐’ฎ๐‘“โˆˆ๐‘›(๐œ†,๐›ผ,๐‘) and ๐‘”โˆˆ๐’ฆ(0), then ||๐‘||ฮฆ(๐‘›,๐œ†,๐›ผ,๐‘)โ‹…(๐‘“โˆ—๐‘”)(๐‘ง)โ‰บ๐‘”(๐‘ง),(3.4)โ„œ(๐‘“)>โˆ’(1โˆ’๐›ผ)+2๐‘›๎€บ||๐‘||๎€ป(1+๐œ†)1+(1โˆ’๐›ผ)2๐‘›๎€บ||๐‘||๎€ป(1+๐œ†)1+(1โˆ’๐›ผ),(3.5) for 0โ‰ฆ๐œ†โ‰ฆ1,0โ‰ฆ๐›ผ<1,๐‘โˆˆโ„‚โงต{0},๐‘›โˆˆโ„•0,(3.6) where, for convenience, 2ฮฆ(๐‘›,๐œ†,๐›ผ,๐‘)โˆถ=๐‘›โˆ’1๎€บ||๐‘||๎€ป(1+๐œ†)1+(1โˆ’๐›ผ)||๐‘||(1โˆ’๐›ผ)+2๐‘›๎€บ||๐‘||๎€ป.(1+๐œ†)1+(1โˆ’๐›ผ)(3.7) The constant factor 2๐‘›โˆ’1๎€บ||๐‘||๎€ป(1+๐œ†)1+(1โˆ’๐›ผ)||๐‘||(1โˆ’๐›ผ)+2๐‘›๎€บ||๐‘||๎€ป(1+๐œ†)1+(1โˆ’๐›ผ)(3.8) in the subordination result (3.4) cannot be replaced by a larger one.

Proof. Let ๎‚‹๐’ฎ๐‘“โˆˆ๐‘›(๐œ†,๐›ผ,๐‘) and suppose that ๐‘”(๐‘ง)=๐‘ง+โˆž๎“๐‘—=2๐‘๐‘—๐‘ง๐‘—โˆˆ๐’ฆโˆถ=๐’ฆ(0),(3.9) then ๎ƒฉฮฆ(๐‘›,๐œ†,๐›ผ,๐‘)โ‹…(๐‘“โˆ—๐‘”)(๐‘ง)=ฮฆ(๐‘›,๐œ†,๐›ผ,๐‘)โ‹…๐‘ง+โˆž๎“๐‘—=2๐‘Ž๐‘—๐‘๐‘—๐‘ง๐‘—๎ƒช,(3.10) where ฮฆ(๐‘›,๐œ†,๐›ผ,๐‘) is defined by (3.7).
If ๎€ฝฮฆ(๐‘›,๐œ†,๐›ผ,๐‘)โ‹…๐‘Ž๐‘—๎€พโˆž๐‘—=1(3.11) is a subordinating factor sequence with ๐‘Ž1=1, then the subordination result (3.4) holds. By Lemma 3.1, we know that this is equivalent to the inequality โ„œ๎ƒฉ1+โˆž๎“๐‘—=12๐‘›๎€บ||๐‘||๎€ป(1+๐œ†)1+(1โˆ’๐›ผ)||๐‘||(1โˆ’๐›ผ)+2๐‘›๎€บ||๐‘||๎€ป๐‘Ž(1+๐œ†)1+(1โˆ’๐›ผ)๐‘—๐‘ง๐‘—๎ƒช>0(๐‘งโˆˆ๐•Œ).(3.12) Since ๐‘—๐‘›๎€บ||๐‘||(1โˆ’๐œ†+๐œ†๐‘—)๐‘—โˆ’1+(1โˆ’๐›ผ)๎€ป๎€ท๐‘—โ‰ง2;0โ‰ฆ๐œ†โ‰ฆ1;0โ‰ฆ๐›ผ<1;๐‘โˆˆโ„‚โงต{0};๐‘›โˆˆโ„•0๎€ธ(3.13) is an increasing function of ๐‘—, and using Theorem 2.1, we have โ„œ๎ƒฉ1+โˆž๎“๐‘—=12๐‘›๎€บ||๐‘||๎€ป(1+๐œ†)1+(1โˆ’๐›ผ)||๐‘||(1โˆ’๐›ผ)+2๐‘›๎€บ||๐‘||๎€ป๐‘Ž(1+๐œ†)1+(1โˆ’๐›ผ)๐‘—๐‘ง๐‘—๎ƒช๎ƒฉ2=โ„œ1+๐‘›(๎€บ||๐‘||(๎€ป1+๐œ†)1+1โˆ’๐›ผ)||๐‘||(1โˆ’๐›ผ)+2๐‘›๎€บ||๐‘||๎€ป๐‘Ž(1+๐œ†)1+(1โˆ’๐›ผ)1๐‘ง+1||๐‘||(1โˆ’๐›ผ)+2๐‘›๎€บ||๐‘||๎€ปโ‹…(1+๐œ†)1+(1โˆ’๐›ผ)โˆž๎“๐‘—=22๐‘›๎€บ||๐‘||๎€ป๐‘Ž(1+๐œ†)1+(1โˆ’๐›ผ)๐‘—๐‘ง๐‘—๎ƒช2โ‰ง1โˆ’๐‘›๎€บ||๐‘||๎€ป(1+๐œ†)1+(1โˆ’๐›ผ)||๐‘||(1โˆ’๐›ผ)+2๐‘›๎€บ||๐‘||๎€ป๐‘Ÿโˆ’1(1+๐œ†)1+(1โˆ’๐›ผ)||๐‘||(1โˆ’๐›ผ)+2๐‘›๎€บ||๐‘||๎€ปโ‹…(1+๐œ†)1+(1โˆ’๐›ผ)โˆž๎“๐‘—=22๐‘›๎€บ||๐‘||๎€ป||๐‘Ž(1+๐œ†)1+(1โˆ’๐›ผ)๐‘—||๐‘Ÿ๐‘—2>1โˆ’๐‘›๎€บ||๐‘||๎€ป(1+๐œ†)1+(1โˆ’๐›ผ)||๐‘||(1โˆ’๐›ผ)+2๐‘›๎€บ||๐‘||๎€ป||๐‘||(1+๐œ†)1+(1โˆ’๐›ผ)๐‘Ÿโˆ’(1โˆ’๐›ผ)||๐‘||(1โˆ’๐›ผ)+2๐‘›๎€บ||๐‘||๎€ป๐‘Ÿ(1+๐œ†)1+(1โˆ’๐›ผ)=1โˆ’๐‘Ÿ>0(|๐‘ง|=๐‘Ÿ<1).(3.14) This evidently proves the inequality (3.12), and hence also the subordination result (3.4), asserted by Theorem 3.2. The inequality (3.5) asserted by Theorem 3.2 follows from (3.4) by setting ๐‘ง๐‘”(๐‘ง)==1โˆ’๐‘งโˆž๎“๐‘—=1๐‘ง๐‘—โˆˆ๐’ฆ.(3.15) Finally, we consider the function ๐‘“0 defined by ๐‘“0||๐‘||(๐‘ง)โˆถ=๐‘งโˆ’(1โˆ’๐›ผ)2๐‘›๎€บ||๐‘||๎€ป๐‘ง(1+๐œ†)1+(1โˆ’๐›ผ)2๎€ท๐‘›โˆˆโ„•0๎€ธ,;0โ‰ฆ๐œ†โ‰ฆ1;0โ‰ฆ๐›ผ<1;๐‘โˆˆโ„‚โงต{0}(3.16) which belongs to the class ๎‚‹๐’ฎ๐‘›(๐œ†,๐›ผ,๐‘). Thus, by (3.4), we know that ฮฆ(๐‘›,๐œ†,๐›ผ,๐‘)โ‹…๐‘“0๐‘ง(๐‘ง)โ‰บ1โˆ’๐‘ง(๐‘งโˆˆ๐•Œ).(3.17) Furthermore, it can be easily verified for the function ๐‘“0 given by (3.16) that min๐‘งโˆˆ๐•Œ๎€ฝโ„œ๎€ทฮฆ(๐‘›,๐œ†,๐›ผ,๐‘)โ‹…๐‘“01(๐‘ง)๎€ธ๎€พ=โˆ’2.(3.18) We thus complete the proof of Theorem 3.2.

The proof of the following subordination result is much akin to that of Theorem 3.2. We, therefore, choose to omit the analogous details involved.

Corollary 3.3. If ๎‚‹โ„ณ๐‘“โˆˆ๐‘›(๐œ†,๐›ผ,๐‘) and ๐‘”โˆˆ๐’ฆ(0), then ||๐‘||(ฮจ(๐‘›,๐œ†,๐›ฝ,๐‘)โ‹…(๐‘“โˆ—๐‘”)(๐‘ง)โ‰บ๐‘”(๐‘ง),(3.19)โ„œ(๐‘“)>โˆ’๐›ฝโˆ’1)+2๐‘›โˆ’1(๎€ท||||||||๎€ธ1+๐œ†)๐‘โˆ’1+2+1โˆ’(2๐›ฝโˆ’1)๐‘2๐‘›โˆ’1๎€ท||||||||๎€ธ(1+๐œ†)๐‘โˆ’1+2+1โˆ’(2๐›ฝโˆ’1)๐‘,(3.20) for 0โ‰ฆ๐œ†โ‰ฆ1,๐›ฝ>1,๐‘โˆˆโ„‚โงต{0},๐‘›โˆˆโ„•0,(3.21) where, for convenience, 2ฮจ(๐‘›,๐œ†,๐›ฝ,๐‘)โˆถ=๐‘›โˆ’2๎€ท||||||||๎€ธ(1+๐œ†)๐‘โˆ’1+2+1โˆ’(2๐›ฝโˆ’1)๐‘||๐‘||(๐›ฝโˆ’1)+2๐‘›โˆ’1๎€ท||||||||๎€ธ(1+๐œ†)๐‘โˆ’1+2+1โˆ’(2๐›ฝโˆ’1)๐‘.(3.22) The constant factor 2๐‘›โˆ’2๎€ท||||||||๎€ธ(1+๐œ†)๐‘โˆ’1+2+1โˆ’(2๐›ฝโˆ’1)๐‘||๐‘||(๐›ฝโˆ’1)+2๐‘›โˆ’1๎€ท||||||||๎€ธ(1+๐œ†)๐‘โˆ’1+2+1โˆ’(2๐›ฝโˆ’1)๐‘(3.23) in the subordination result (3.19) cannot be replaced by a larger one.

Remark 3.4. Putting ๐œ†=0, ๐‘=1, and ๐‘›=0or1 in Corollary 3.3, we get the corresponding results obtained by Srivastava and Attiya [4].

Acknowledgments

The present investigation was supported by the National Natural Science Foundation under grants 11101053, 70971013, and 71171024, the Natural Science Foundation of Hunan Province under grant 09JJ1010, the Key Project of Chinese Ministry of Education under grant 211118, the Excellent Youth Foundation of Educational Committee of Hunan Province under grant 10B002, the Open Fund Project of Key Research Institute of Philosophies and Social Sciences in Hunan Universities under grant 11FEFM02, and the Key Project of Natural Science Foundation of Educational Committee of Henan Province under grant 12A110002 of China.