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Abstract and Applied Analysis
VolumeΒ 2011, Article IDΒ 456729, 16 pages
http://dx.doi.org/10.1155/2011/456729
Research Article

A Class of Analytic Functions with Missing Coefficients

1Department of Mathematics, Soochow University, Jiangsu, Suzhou 215006, China
2Department of Mathematics, Yangzhou University, Jiangsu, Yangzhou 225002, China

Received 1 March 2011; Accepted 9 May 2011

Academic Editor: PaulΒ Eloe

Copyright Β© 2011 Ding-Gong Yang and Jin-Lin Liu. 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

Let 𝑇 𝑛 ( 𝐴 , 𝐡 , 𝛾 , 𝛼 ) ( βˆ’ 1 ≀ 𝐡 < 1 , 𝐡 < 𝐴 , 0 < 𝛾 ≀ 1 and 𝛼 > 0 ) denote the class of functions of the form βˆ‘ 𝑓 ( 𝑧 ) = 𝑧 + ∞ π‘˜ = 𝑛 + 1 π‘Ž π‘˜ 𝑧 π‘˜ ( 𝑛 ∈ 𝑁 = { 1 , 2 , 3 , … } ) , which are analytic in the open unit disk π‘ˆ and satisfy the following subordination condition 𝑓 ξ…ž ( 𝑧 ) + 𝛼 𝑧 𝑓 ξ…ž ξ…ž ( 𝑧 ) β‰Ί ( ( 1 + 𝐴 𝑧 ) / ( 1 + 𝐡 𝑧 ) ) 𝛾 , for ( 𝑧 ∈ π‘ˆ ; 𝐴 ≀ 1 ; 0 < 𝛾 < 1 ) , ( 1 + 𝐴 𝑧 ) / ( 1 + 𝐡 𝑧 ) , for ( 𝑧 ∈ π‘ˆ ; 𝛾 = 1 ) . We obtain sharp bounds on R e 𝑓 ξ…ž ( 𝑧 ) , R e 𝑓 ( 𝑧 ) / 𝑧 , | 𝑓 ( 𝑧 ) | , and coefficient estimates for functions 𝑓 ( 𝑧 ) belonging to the class 𝑇 𝑛 ( 𝐴 , 𝐡 , 𝛾 , 𝛼 ) . Conditions for univalency and starlikeness, convolution properties, and the radius of convexity are also considered.

1. Introduction

Let 𝐴 𝑛 denote the class of functions of the form 𝑓 ( 𝑧 ) = 𝑧 + ∞  π‘˜ = 𝑛 + 1 π‘Ž π‘˜ 𝑧 π‘˜ ( 𝑛 ∈ 𝑁 = { 1 , 2 , 3 , … } ) , ( 1 . 1 ) which are analytic in the open unit disk π‘ˆ = { 𝑧 ∢ 𝑧 ∈ 𝐢 a n d | 𝑧 | < 1 } . Let 𝑆 𝑛 and 𝑆 βˆ— 𝑛 denote the subclasses of 𝐴 𝑛 whose members are univalent and starlike, respectively.

For functions 𝑓 ( 𝑧 ) and 𝑔 ( 𝑧 ) analytic in π‘ˆ , we say that 𝑓 ( 𝑧 ) is subordinate to 𝑔 ( 𝑧 ) in π‘ˆ and we write 𝑓 ( 𝑧 ) β‰Ί 𝑔 ( 𝑧 ) ( 𝑧 ∈ π‘ˆ ) , if there exists an analytic function 𝑀 ( 𝑧 ) in π‘ˆ such that | | | | 𝑀 ( 𝑧 ) ≀ | 𝑧 | , 𝑓 ( 𝑧 ) = 𝑔 ( 𝑀 ( 𝑧 ) ) ( 𝑧 ∈ π‘ˆ ) . ( 1 . 2 ) Furthermore, if the function 𝑔 ( 𝑧 ) is univalent in π‘ˆ , then 𝑓 ( 𝑧 ) β‰Ί 𝑔 ( 𝑧 ) ( 𝑧 ∈ π‘ˆ ) ⟺ 𝑓 ( 0 ) = 𝑔 ( 0 ) , 𝑓 ( π‘ˆ ) βŠ‚ 𝑔 ( π‘ˆ ) . ( 1 . 3 ) Throughout our present discussion, we assume that 𝑛 ∈ 𝑁 , βˆ’ 1 ≀ 𝐡 < 1 , 𝐡 < 𝐴 , 𝛼 > 0 , 𝛽 < 1 , 0 < 𝛾 ≀ 1 . ( 1 . 4 ) We introduce the following subclass of 𝐴 𝑛 .

Definition 1.1. A function 𝑓 ( 𝑧 ) ∈ 𝐴 𝑛 is said to be in the class 𝑇 𝑛 ( 𝐴 , 𝐡 , 𝛾 , 𝛼 ) if it satisfies 𝑓 ξ…ž ( 𝑧 ) + 𝛼 𝑧 𝑓 ξ…ž ξ…ž ( 𝑧 ) β‰Ί β„Ž ( 𝑧 ) ( 𝑧 ∈ π‘ˆ ) , ( 1 . 5 ) where ⎧ βŽͺ ⎨ βŽͺ ⎩ ξ‚€ β„Ž ( 𝑧 ) = 1 + 𝐴 𝑧  1 + 𝐡 𝑧 𝛾 , ( 𝐴 ≀ 1 ; 0 < 𝛾 < 1 ) , 1 + 𝐴 𝑧 1 + 𝐡 𝑧 , ( 𝛾 = 1 ) . ( 1 . 6 ) The classes 𝑇 1 ( 1 βˆ’ 2 𝛽 , βˆ’ 1 , 1 , 1 ) = 𝑅 ( 𝛽 ) ( 𝛽 = 0 o r 𝛽 < 1 ) , 𝑇 1  𝑅 ( 𝐴 , 0 , 1 , 𝛼 ) = ( 𝛼 , 𝐴 ) ( 𝐴 > 0 ) ( 1 . 7 ) have been studied by several authors (see [15]). Recently, Gao and Zhou [6] showed some mapping properties of the following subclass of 𝐴 1 : 𝑅 ξ€½ 𝑓 ( 𝛽 , 𝛼 ) = ( 𝑧 ) ∈ 𝐴 1 ξ€½ 𝑓 ∢ R e ξ…ž ( 𝑧 ) + 𝛼 𝑧 𝑓 ξ…ž ξ…ž ξ€Ύ ξ€Ύ ( 𝑧 ) > 𝛽 ( 𝑧 ∈ π‘ˆ ) . ( 1 . 8 ) Note that 𝑅 ( 𝛽 , 1 ) = 𝑅 ( 𝛽 ) , 𝑇 1 ( 1 βˆ’ 2 𝛽 , βˆ’ 1 , 1 , 𝛼 ) = 𝑅 ( 𝛽 , 𝛼 ) . ( 1 . 9 ) For further information of the above classes (with 𝛾 = 1 ) and related analytic function classes, see Srivastava et al. [7], Yang and Liu [8], Kim [9], and Kim and Srivastava [10].

In this paper, we obtain sharp bounds on R e 𝑓 ξ…ž ( 𝑧 ) , R e ( 𝑓 ( 𝑧 ) / 𝑧 ) , | 𝑓 ( 𝑧 ) | , and coefficient estimates for functions 𝑓 ( 𝑧 ) belonging to the class 𝑇 𝑛 ( 𝐴 , 𝐡 , 𝛾 , 𝛼 ) . Conditions for univalency and starlikeness, convolution properties, and the radius of convexity are also presented. One can see that the methods used in [6] do not work for the more general class 𝑇 𝑛 ( 𝐴 , 𝐡 , 𝛾 , 𝛼 ) than 𝑅 ( 𝛽 , 𝛼 ) .

2. The bounds on R e 𝑓 ξ…ž ( 𝑧 ) , R e ( 𝑓 ( 𝑧 ) / 𝑧 ) , and | 𝑓 ( 𝑧 ) | in 𝑇 𝑛 ( 𝐴 , 𝐡 , 𝛾 , 𝛼 )

In this section, we let πœ† π‘š ⎧ βŽͺ ⎨ βŽͺ ⎩ ( 𝐴 , 𝐡 , 𝛾 ) = π‘š  𝑗 = 0  𝛾 𝑗 ξƒͺ 𝐴 ξƒͺ  βˆ’ 𝛾 π‘š βˆ’ 𝑗 𝑗 𝐡 π‘š βˆ’ 𝑗 , ( 𝐴 ≀ 1 ; 0 < 𝛾 < 1 ) , ( 𝐴 βˆ’ 𝐡 ) ( βˆ’ 𝐡 ) π‘š βˆ’ 1 , ( 𝛾 = 1 ) , ( 2 . 1 ) where π‘š ∈ 𝑁 and  𝛾 𝑗 ξƒͺ = ⎧ βŽͺ ⎨ βŽͺ ⎩ 𝛾 ( 𝛾 βˆ’ 1 ) β‹― ( 𝛾 βˆ’ 𝑗 + 1 ) 𝑗 ! , ( 𝑗 = 1 , 2 , … , π‘š ) , 1 , ( 𝑗 = 0 ) . ( 2 . 2 ) With (2.1), it is easily seen that the function β„Ž ( 𝑧 ) given by (1.6) can be expressed as β„Ž ( 𝑧 ) = 1 + ∞  π‘š = 1 πœ† π‘š ( 𝐴 , 𝐡 , 𝛾 ) 𝑧 π‘š ( 𝑧 ∈ π‘ˆ ) . ( 2 . 3 )

Theorem 2.1. Let 𝑓 ( 𝑧 ) ∈ 𝑇 𝑛 ( 𝐴 , 𝐡 , 𝛾 , 𝛼 ) . Then, for | 𝑧 | = π‘Ÿ < 1 , R e 𝑓 ξ…ž ( 𝑧 ) β‰₯ 1 + ∞  π‘š = 1 ( βˆ’ 1 ) π‘š πœ† π‘š ( 𝐴 , 𝐡 , 𝛾 ) π‘Ÿ 𝛼 𝑛 π‘š + 1 𝑛 π‘š , R e 𝑓 ξ…ž ( 𝑧 ) ≀ 1 + ∞  π‘š = 1 πœ† π‘š ( 𝐴 , 𝐡 , 𝛾 ) π‘Ÿ 𝛼 𝑛 π‘š + 1 𝑛 π‘š . ( 2 . 4 ) The bounds in (2.4) are sharp for the function 𝑓 𝑛 ( 𝑧 ) defined by 𝑓 𝑛 ( 𝑧 ) = 𝑧 + ∞  π‘š = 1 πœ† π‘š ( 𝐴 , 𝐡 , 𝛾 ) 𝑧 ( 𝑛 π‘š + 1 ) ( 𝛼 𝑛 π‘š + 1 ) 𝑛 π‘š + 1 ( 𝑧 ∈ π‘ˆ ) . ( 2 . 5 )

Proof. The analytic function β„Ž ( 𝑧 ) given by (1.6) is convex (univalent) in π‘ˆ (cf. [11]) and satisfies β„Ž ( 𝑧 ) = β„Ž ( 𝑧 ) ( 𝑧 ∈ π‘ˆ ) . Thus, for | 𝜁 | ≀ 𝜎 ( 𝜁 ∈ 𝐢 a n d 𝜎 < 1 ) , β„Ž ( βˆ’ 𝜎 ) ≀ R e β„Ž ( 𝜁 ) ≀ β„Ž ( 𝜎 ) . ( 2 . 6 )
Let 𝑓 ( 𝑧 ) ∈ 𝑇 𝑛 ( 𝐴 , 𝐡 , 𝛾 , 𝛼 ) . Then, we can write 𝑓 ξ…ž ( 𝑧 ) + 𝛼 𝑧 𝑓 ξ…ž ξ…ž ( 𝑧 ) = β„Ž ( 𝑀 ( 𝑧 ) ) ( 𝑧 ∈ π‘ˆ ) , ( 2 . 7 ) where 𝑀 ( 𝑧 ) = 𝑀 𝑛 𝑧 𝑛 + 𝑀 𝑛 + 1 𝑧 𝑛 + 1 + β‹― is analytic and | 𝑀 ( 𝑧 ) | < 1 for 𝑧 ∈ π‘ˆ . By the Schwarz lemma, we know that | 𝑀 ( 𝑧 ) | ≀ | 𝑧 | 𝑛 ( 𝑧 ∈ π‘ˆ ) . It follows from (2.7) that ξ€· 𝑧 1 / 𝛼 𝑓 ξ…ž ξ€Έ ( 𝑧 ) ξ…ž = 1 𝛼 𝑧 ( 1 / 𝛼 ) βˆ’ 1 β„Ž ( 𝑀 ( 𝑧 ) ) , ( 2 . 8 ) which leads to 𝑓 ξ…ž 1 ( 𝑧 ) = 𝛼 𝑧 βˆ’ 1 / 𝛼 ξ€œ 𝑧 0 𝜁 ( 1 / 𝛼 ) βˆ’ 1 β„Ž ( 𝑀 ( 𝜁 ) ) 𝑑 𝜁 ( 2 . 9 ) or to 𝑓 ξ…ž 1 ( 𝑧 ) = 𝛼 ξ€œ 1 0 𝑑 ( 1 / 𝛼 ) βˆ’ 1 β„Ž ( 𝑀 ( 𝑑 𝑧 ) ) 𝑑 𝑑 ( 𝑧 ∈ π‘ˆ ) . ( 2 . 1 0 ) Since | | | | 𝑀 ( 𝑑 𝑧 ) ≀ ( 𝑑 π‘Ÿ ) 𝑛 ( | 𝑧 | = π‘Ÿ < 1 ; 0 ≀ 𝑑 ≀ 1 ) , ( 2 . 1 1 ) we deduce from (2.6) and (2.10) that 1 𝛼 ξ€œ 1 0 𝑑 ( 1 / 𝛼 ) βˆ’ 1 β„Ž ( βˆ’ ( 𝑑 π‘Ÿ ) 𝑛 ) 𝑑 𝑑 ≀ R e 𝑓 ξ…ž 1 ( 𝑧 ) ≀ 𝛼 ξ€œ 1 0 𝑑 ( 1 / 𝛼 ) βˆ’ 1 β„Ž ( ( 𝑑 π‘Ÿ ) 𝑛 ) 𝑑 𝑑 . ( 2 . 1 2 ) Now, by using (2.3) and (2.12), we can obtain (2.4).
Furthermore, for the function 𝑓 𝑛 ( 𝑧 ) defined by (2.5), we find that 𝑓 ξ…ž 𝑛 ( 𝑧 ) = 1 + ∞  π‘š = 1 πœ† π‘š ( 𝐴 , 𝐡 , 𝛾 ) 𝑧 𝛼 𝑛 π‘š + 1 𝑛 π‘š , 𝑓 ( 2 . 1 3 ) ξ…ž 𝑛 ( 𝑧 ) + 𝛼 𝑧 𝑓 𝑛 ξ…ž ξ…ž ( 𝑧 ) = 1 + ∞  π‘š = 1 πœ† π‘š ( 𝐴 , 𝐡 , 𝛾 ) 𝑧 𝑛 π‘š = β„Ž ( 𝑧 𝑛 ) β‰Ί β„Ž ( 𝑧 ) ( 𝑧 ∈ π‘ˆ ) . ( 2 . 1 4 ) Hence, 𝑓 𝑛 ( 𝑧 ) ∈ 𝑇 𝑛 ( 𝐴 , 𝐡 , 𝛾 , 𝛼 ) and from (2.13), we see that the bounds in (2.4) are the best possible.
Hereafter, we write 𝑇 𝑛 ( 𝐴 , 𝐡 , 1 , 𝛼 ) = 𝑇 𝑛 ( 𝐴 , 𝐡 , 𝛼 ) . ( 2 . 1 5 )

Corollary 2.2. Let 𝑓 ( 𝑧 ) ∈ 𝑇 𝑛 ( 𝐴 , 𝐡 , 𝛼 ) . Then, for 𝑧 ∈ π‘ˆ , R e 𝑓 ξ…ž ( 𝑧 ) > 1 βˆ’ ( 𝐴 βˆ’ 𝐡 ) ∞  π‘š = 1 𝐡 π‘š βˆ’ 1 , 𝛼 𝑛 π‘š + 1 ( 2 . 1 6 ) R e 𝑓 ξ…ž ( 𝑧 ) < 1 + ( 𝐴 βˆ’ 𝐡 ) ∞  π‘š = 1 ( βˆ’ 𝐡 ) π‘š βˆ’ 1 𝛼 𝑛 π‘š + 1 ( 𝐡 β‰  βˆ’ 1 ) . ( 2 . 1 7 ) The results are sharp.

Proof. For 𝛾 = 1 , it follows from (2.12) (used in the proof of Theorem 2.1) that R e 𝑓 ξ…ž 1 ( 𝑧 ) > 𝛼 ξ€œ 1 0 𝑑 ( 1 / 𝛼 ) βˆ’ 1 ξ‚΅ 1 βˆ’ 𝐴 𝑑 𝑛 1 βˆ’ 𝐡 𝑑 𝑛 ξ‚Ά 𝑑 𝑑 , R e 𝑓 ξ…ž 1 ( 𝑧 ) < 𝛼 ξ€œ 1 0 𝑑 ( 1 / 𝛼 ) βˆ’ 1 ξ‚΅ 1 + 𝐴 𝑑 𝑛 1 + 𝐡 𝑑 𝑛 ξ‚Ά 𝑑 𝑑 ( 𝐡 β‰  βˆ’ 1 ) , ( 2 . 1 8 ) for 𝑧 ∈ π‘ˆ . From these, we have the desired results.

The bounds in (2.16) and (2.17) are sharp for the function 𝑓 𝑛 ( 𝑧 ) = 𝑧 + ( 𝐴 βˆ’ 𝐡 ) ∞  π‘š = 1 ( βˆ’ 𝐡 ) π‘š βˆ’ 1 𝑧 ( 𝑛 π‘š + 1 ) ( 𝛼 𝑛 π‘š + 1 ) 𝑛 π‘š + 1 ∈ 𝑇 𝑛 ( 𝐴 , 𝐡 , 𝛼 ) . ( 2 . 1 9 )

Theorem 2.3. Let 𝑓 ( 𝑧 ) ∈ 𝑇 𝑛 ( 𝐴 , 𝐡 , 𝛾 , 𝛼 ) . Then, for | 𝑧 | = π‘Ÿ < 1 , R e 𝑓 ( 𝑧 ) 𝑧 β‰₯ 1 + ∞  π‘š = 1 ( βˆ’ 1 ) π‘š πœ† π‘š ( 𝐴 , 𝐡 , 𝛾 ) π‘Ÿ ( 𝑛 π‘š + 1 ) ( 𝛼 𝑛 π‘š + 1 ) 𝑛 π‘š , R e 𝑓 ( 𝑧 ) 𝑧 ≀ 1 + ∞  π‘š = 1 πœ† π‘š ( 𝐴 , 𝐡 , 𝛾 ) π‘Ÿ ( 𝑛 π‘š + 1 ) ( 𝛼 𝑛 π‘š + 1 ) 𝑛 π‘š . ( 2 . 2 0 ) The results are sharp.

Proof. Noting that ξ€œ 𝑓 ( 𝑧 ) = 𝑧 1 0 𝑓 ξ…ž ( 𝑒 𝑧 ) 𝑑 𝑒 , R e 𝑓 ( 𝑧 ) 𝑧 = ξ€œ 1 0 R e 𝑓 ξ…ž ( 𝑒 𝑧 ) 𝑑 𝑒 ( 𝑧 ∈ π‘ˆ ) , ( 2 . 2 1 ) an application of Theorem 2.1 yields (2.20). Furthermore, the results are sharp for the function 𝑓 𝑛 ( 𝑧 ) defined by (2.5).

Corollary 2.4. Let 𝑓 ( 𝑧 ) ∈ 𝑇 𝑛 ( 𝐴 , 𝐡 , 𝛼 ) . Then, for 𝑧 ∈ π‘ˆ , R e 𝑓 ( 𝑧 ) 𝑧 > 1 βˆ’ ( 𝐴 βˆ’ 𝐡 ) ∞  π‘š = 1 𝐡 π‘š βˆ’ 1 , ( 𝑛 π‘š + 1 ) ( 𝛼 𝑛 π‘š + 1 ) R e 𝑓 ( 𝑧 ) 𝑧 < 1 + ( 𝐴 βˆ’ 𝐡 ) ∞  π‘š = 1 ( βˆ’ 𝐡 ) π‘š βˆ’ 1 . ( 𝑛 π‘š + 1 ) ( 𝛼 𝑛 π‘š + 1 ) ( 2 . 2 2 ) The results are sharp for the function 𝑓 𝑛 ( 𝑧 ) defined by (2.19).

Proof. For 𝑓 ( 𝑧 ) ∈ 𝑇 𝑛 ( 𝐴 , 𝐡 , 𝛼 ) , it follows from (2.6) and (2.10) (with 𝛾 = 1 ) that 1 𝛼 ξ€œ 1 0 𝑑 ( 1 / 𝛼 ) βˆ’ 1 ξ‚΅ 1 βˆ’ 𝐴 ( 𝑒 𝑑 ) 𝑛 1 βˆ’ 𝐡 ( 𝑒 𝑑 ) 𝑛 ξ‚Ά 𝑑 𝑑 < R e 𝑓 ξ…ž 1 ( 𝑒 𝑧 ) < 𝛼 ξ€œ 1 0 𝑑 ( 1 / 𝛼 ) βˆ’ 1 ξ‚΅ 1 + 𝐴 ( 𝑒 𝑑 ) 𝑛 1 + 𝐡 ( 𝑒 𝑑 ) 𝑛 ξ‚Ά 𝑑 𝑑 , ( 2 . 2 3 ) for 𝑧 ∈ π‘ˆ and 0 < 𝑒 ≀ 1 . Making use of (2.21) and (2.23), we can obtain (2.22).

Theorem 2.5. Let 𝑓 ( 𝑧 ) ∈ 𝑇 1 ( 𝐴 , 𝐡 , 𝛼 ) and 𝑔 ( 𝑧 ) ∈ 𝑇 1 ( 𝐴 0 , 𝐡 0 , 𝛼 0 ) ( βˆ’ 1 ≀ 𝐡 0 < 1 , 𝐡 0 < 𝐴 0 and 𝛼 0 > 0 ). If ξ€· 𝐴 0 βˆ’ 𝐡 0 ξ€Έ ∞  π‘š = 1 𝐡 0 π‘š βˆ’ 1 ξ€· 𝛼 ( π‘š + 1 ) 0 ξ€Έ ≀ 1 π‘š + 1 2 , ( 2 . 2 4 ) then ( 𝑓 βˆ— 𝑔 ) ( 𝑧 ) ∈ 𝑇 1 ( 𝐴 , 𝐡 , 𝛼 ) , where the symbol βˆ— stands for the familiar Hadamard product (or convolution) of two analytic functions in π‘ˆ .

Proof. Since 𝑔 ( 𝑧 ) ∈ 𝑇 1 ( 𝐴 0 , 𝐡 0 , 𝛼 0 ) ( βˆ’ 1 ≀ 𝐡 0 < 1 , 𝐡 0 < 𝐴 0 and 𝛼 0 > 0 ), it follows from Corollary 2.4 (with 𝑛 = 1 ) and (2.24) that R e 𝑔 ( 𝑧 ) 𝑧 ξ€· 𝐴 > 1 βˆ’ 0 βˆ’ 𝐡 0 ξ€Έ ∞  π‘š = 1 𝐡 0 π‘š βˆ’ 1 ξ€· 𝛼 ( π‘š + 1 ) 0 ξ€Έ β‰₯ 1 π‘š + 1 2 ( 𝑧 ∈ π‘ˆ ) . ( 2 . 2 5 ) Thus, 𝑔 ( 𝑧 ) / 𝑧 has the Herglotz representation 𝑔 ( 𝑧 ) 𝑧 = ξ€œ | π‘₯ | = 1 𝑑 πœ‡ ( π‘₯ ) ( 1 βˆ’ π‘₯ 𝑧 𝑧 ∈ π‘ˆ ) , ( 2 . 2 6 ) where πœ‡ ( π‘₯ ) is a probability measure on the unit circle | π‘₯ | = 1 and ∫ | π‘₯ | = 1 𝑑 πœ‡ ( π‘₯ ) = 1 .
For 𝑓 ( 𝑧 ) ∈ 𝑇 1 ( 𝐴 , 𝐡 , 𝛼 ) , we have ( 𝑓 βˆ— 𝑔 ) ξ…ž ( 𝑧 ) + 𝛼 𝑧 ( 𝑓 βˆ— 𝑔 ) ξ…ž ξ…ž ( 𝑧 ) = 𝐹 ( 𝑧 ) βˆ— 𝑔 ( 𝑧 ) 𝑧 ( 𝑧 ∈ π‘ˆ ) , ( 2 . 2 7 ) where 𝐹 ( 𝑧 ) = 𝑓 ξ…ž ( 𝑧 ) + 𝛼 𝑧 𝑓 ξ…ž ξ…ž ( 𝑧 ) β‰Ί 1 + 𝐴 𝑧 1 + 𝐡 𝑧 ( 𝑧 ∈ π‘ˆ ) . ( 2 . 2 8 ) In view of the function ( 1 + 𝐴 𝑧 ) / ( 1 + 𝐡 𝑧 ) is convex (univalent) in π‘ˆ , we deduce from (2.26) to (2.28) that ( 𝑓 βˆ— 𝑔 ) ξ…ž ( 𝑧 ) + 𝛼 𝑧 ( 𝑓 βˆ— 𝑔 ) ξ…ž ξ…ž ( ξ€œ 𝑧 ) = | π‘₯ | = 1 𝐹 ( π‘₯ 𝑧 ) 𝑑 πœ‡ ( π‘₯ ) β‰Ί 1 + 𝐴 𝑧 ( 1 + 𝐡 𝑧 𝑧 ∈ π‘ˆ ) . ( 2 . 2 9 ) This shows that ( 𝑓 βˆ— 𝑔 ) ( 𝑧 ) ∈ 𝑇 1 ( 𝐴 , 𝐡 , 𝛼 ) .

Corollary 2.6. Let 𝑓 ( 𝑧 ) ∈ 𝑇 1 ( 𝐴 , 𝐡 , 𝛼 ) , 𝑔 ( 𝑧 ) ∈ 𝑅 ( 𝛽 , 1 ) and πœ‹ 𝛽 β‰₯ βˆ’ 2 βˆ’ 9 1 2 βˆ’ πœ‹ 2 . ( 2 . 3 0 ) Then, ( 𝑓 βˆ— 𝑔 ) ( 𝑧 ) ∈ 𝑇 1 ( 𝐴 , 𝐡 , 𝛼 ) .

Proof. By taking 𝐴 0 = 1 βˆ’ 2 𝛽 , 𝐡 0 = βˆ’ 1 and 𝛼 0 = 1 , (2.24) in Theorem 2.5 becomes 2 ( 1 βˆ’ 𝛽 ) ∞  π‘š = 1 ( βˆ’ 1 ) π‘š βˆ’ 1 ( π‘š + 1 ) 2 ξ‚΅ πœ‹ = 2 ( 1 βˆ’ 𝛽 ) 1 βˆ’ 2 ξ‚Ά ≀ 1 1 2 2 , ( 2 . 3 1 ) that is, πœ‹ 𝛽 β‰₯ βˆ’ 2 βˆ’ 9 1 2 βˆ’ πœ‹ 2 . ( 2 . 3 2 ) Hence, the desired result follows as a special case from Theorem 2.5.

Remark 2.7. R. Singh and S. Singh [4, Theorem 3] proved that, if 𝑓 ( 𝑧 ) and 𝑔 ( 𝑧 ) belong to 𝑅 ( 0 , 1 ) , then ( 𝑓 βˆ— 𝑔 ) ( 𝑧 ) ∈ 𝑅 ( 0 , 1 ) . Obviously, for βˆ’ πœ‹ 2 βˆ’ 9 1 2 βˆ’ πœ‹ 2 ≀ 𝛽 < 0 , ( 2 . 3 3 ) Corollary 2.6 generalizes and improves Theorem 3 in [4].

Theorem 2.8. Let 𝑓 ( 𝑧 ) ∈ 𝑇 𝑛 ( 𝐴 , 𝐡 , 𝛾 , 𝛼 ) and 𝐴 𝐡 ≀ 1 . Then, for | 𝑧 | = π‘Ÿ < 1 , | | | | 𝑓 ( 𝑧 ) ≀ π‘Ÿ + ∞  π‘š = 1 πœ† π‘š ( 𝐴 , 𝐡 , 𝛾 ) π‘Ÿ ( 𝑛 π‘š + 1 ) ( 𝛼 𝑛 π‘š + 1 ) 𝑛 π‘š + 1 . ( 2 . 3 4 ) The result is sharp, with the extremal function 𝑓 𝑛 ( 𝑧 ) defined by (2.5).

Proof. It is well known that for 𝜁 ∈ 𝐢 and | 𝜁 | ≀ 𝜎 < 1 , | | | | 1 + 𝐴 𝜁 βˆ’ 1 + 𝐡 𝜁 1 βˆ’ 𝐴 𝐡 𝜎 2 1 βˆ’ 𝐡 2 𝜎 2 | | | | ≀ ( 𝐴 βˆ’ 𝐡 ) 𝜎 1 βˆ’ 𝐡 2 𝜎 2 . ( 2 . 3 5 ) Since 𝐴 𝐡 ≀ 1 , we have 1 βˆ’ 𝐴 𝐡 𝜎 2 > 0 and so (2.35) leads to | | | | 1 + 𝐴 𝜁 | | | | 1 + 𝐡 𝜁 𝛾 ≀  | | | | 1 βˆ’ 𝐴 𝐡 𝜎 2 1 βˆ’ 𝐡 2 𝜎 2 | | | | + ( 𝐴 βˆ’ 𝐡 ) 𝜎 1 βˆ’ 𝐡 2 𝜎 2 ξƒͺ 𝛾 = ξ‚€ 1 + 𝐴 𝜎  1 + 𝐡 𝜎 𝛾 ξ€· | | 𝜁 | | ξ€Έ . ≀ 𝜎 < 1 ( 2 . 3 6 ) By virtue of (1.6), (2.10), and (2.36), we have | | 𝑓 ξ…ž | | ≀ 1 ( 𝑒 𝑧 ) 𝛼 ξ€œ 1 0 𝑑 ( 1 / 𝛼 ) βˆ’ 1 | | | | 1 β„Ž ( 𝑀 ( 𝑒 𝑑 𝑧 ) ) 𝑑 𝑑 ≀ 𝛼 ξ€œ 1 0 𝑑 ( 1 / 𝛼 ) βˆ’ 1 β„Ž ( ( 𝑒 𝑑 | 𝑧 | ) 𝑛 ) 𝑑 𝑑 , ( 2 . 3 7 ) for 𝑧 ∈ π‘ˆ and 0 ≀ 𝑒 ≀ 1 . Now, by using (2.3), (2.21) and (2.37), we can obtain (2.34).

Theorem 2.9. Let 𝑓 ( 𝑧 ) = 𝑧 + ∞  π‘˜ = 𝑛 + 1 π‘Ž π‘˜ 𝑧 π‘˜ ∈ 𝑇 𝑛 ( 𝐴 , 𝐡 , 𝛾 , 𝛼 ) . ( 2 . 3 8 ) Then, | | π‘Ž π‘˜ | | ≀ 𝛾 ( 𝐴 βˆ’ 𝐡 ) ( π‘˜ ( 𝛼 ( π‘˜ βˆ’ 1 ) + 1 ) π‘˜ β‰₯ 𝑛 + 1 ) . ( 2 . 3 9 ) The result is sharp for each π‘˜ β‰₯ 𝑛 + 1 .

Proof. It is known (cf. [12]) that, if πœ‘ ( 𝑧 ) = ∞  π‘˜ = 1 𝑏 π‘˜ 𝑧 π‘˜ β‰Ί πœ“ ( 𝑧 ) ( 𝑧 ∈ π‘ˆ ) , ( 2 . 4 0 ) where πœ‘ ( 𝑧 ) is analytic in π‘ˆ and πœ“ ( 𝑧 ) = 𝑧 + β‹― is analytic and convex univalent in π‘ˆ , then | 𝑏 π‘˜ | ≀ 1 ( π‘˜ ∈ 𝑁 ) .
By (2.38), we have 𝑓 ξ…ž ( 𝑧 ) + 𝛼 𝑧 𝑓 ξ…ž ξ…ž ( 𝑧 ) βˆ’ 1 = 𝛾 ( 𝐴 βˆ’ 𝐡 ) ∞  π‘˜ = 𝑛 + 1 π‘˜ ( 𝛼 ( π‘˜ βˆ’ 1 ) + 1 ) π‘Ž 𝛾 ( 𝐴 βˆ’ 𝐡 ) π‘˜ 𝑧 π‘˜ βˆ’ 1 β‰Ί πœ“ ( 𝑧 ) ( 𝑧 ∈ π‘ˆ ) , ( 2 . 4 1 ) where πœ“ ( 𝑧 ) = β„Ž ( 𝑧 ) βˆ’ 1 𝛾 ( 𝐴 βˆ’ 𝐡 ) = 𝑧 + β‹― ( 2 . 4 2 ) and β„Ž ( 𝑧 ) is given by (1.6). Since the function πœ“ ( 𝑧 ) is analytic and convex univalent in π‘ˆ , it follows from (2.41) that π‘˜ ( 𝛼 ( π‘˜ βˆ’ 1 ) + 1 ) | | π‘Ž 𝛾 ( 𝐴 βˆ’ 𝐡 ) π‘˜ | | ≀ 1 ( π‘˜ β‰₯ 𝑛 + 1 ) , ( 2 . 4 3 ) which gives (2.39).
Next, we consider the function 𝑓 π‘˜ βˆ’ 1 ( 𝑧 ) = 𝑧 + ∞  π‘š = 1 πœ† π‘š ( 𝐴 , 𝐡 , 𝛾 ) 𝑧 ( π‘š ( π‘˜ βˆ’ 1 ) + 1 ) ( 𝛼 π‘š ( π‘˜ βˆ’ 1 ) + 1 ) π‘š ( π‘˜ βˆ’ 1 ) + 1 ( 𝑧 ∈ π‘ˆ ; π‘˜ β‰₯ 𝑛 + 1 ) . ( 2 . 4 4 ) It is easy to verify that 𝑓 ξ…ž π‘˜ βˆ’ 1 ( 𝑧 ) + 𝛼 𝑧 𝑓 ξ…ž ξ…ž π‘˜ βˆ’ 1 ξ€· 𝑧 ( 𝑧 ) = β„Ž π‘˜ βˆ’ 1 ξ€Έ 𝑓 β‰Ί β„Ž ( 𝑧 ) ( 𝑧 ∈ π‘ˆ ) , π‘˜ βˆ’ 1 𝛾 ( 𝑧 ) = 𝑧 + ( 𝐴 βˆ’ 𝐡 ) 𝑧 π‘˜ ( 𝛼 ( π‘˜ βˆ’ 1 ) + 1 ) π‘˜ + β‹― . ( 2 . 4 5 ) The proof of Theorem 2.9 is completed.

3. The Univalency and Starlikeness of 𝑇 𝑛 ( 𝐴 , 𝐡 , 𝛼 )

Theorem 3.1. 𝑇 𝑛 ( 𝐴 , 𝐡 , 𝛼 ) βŠ‚ 𝑆 𝑛 if and only if ( 𝐴 βˆ’ 𝐡 ) ∞  π‘š = 1 𝐡 π‘š βˆ’ 1 𝛼 𝑛 π‘š + 1 ≀ 1 . ( 3 . 1 )

Proof. Let 𝑓 ( 𝑧 ) ∈ 𝑇 𝑛 ( 𝐴 , 𝐡 , 𝛼 ) and (3.1) be satisfied. Then, by (2.16) in Corollary 2.2, we see that R e 𝑓 β€² ( 𝑧 ) > 0 ( 𝑧 ∈ π‘ˆ ) . Thus, 𝑓 ( 𝑧 ) is close-to-convex and univalent in π‘ˆ .
On the other hand, if ( 𝐴 βˆ’ 𝐡 ) ∞  π‘š = 1 𝐡 π‘š βˆ’ 1 𝛼 𝑛 π‘š + 1 > 1 , ( 3 . 2 ) then the function 𝑓 𝑛 ( 𝑧 ) defined by (2.19) satisfies 𝑓 ξ…ž 𝑛 ( 0 ) = 1 > 0 and 𝑓 ξ…ž 𝑛 ξ€· π‘Ÿ 𝑒 πœ‹ 𝑖 / 𝑛 ξ€Έ = 1 βˆ’ ( 𝐴 βˆ’ 𝐡 ) ∞  π‘š = 1 𝐡 π‘š βˆ’ 1 π‘Ÿ 𝛼 𝑛 π‘š + 1 𝑛 π‘š ⟢ 1 βˆ’ ( 𝐴 βˆ’ 𝐡 ) ∞  π‘š = 1 𝐡 π‘š βˆ’ 1 𝛼 𝑛 π‘š + 1 < 0 ( 3 . 3 ) as π‘Ÿ β†’ 1 . Hence, there exists a point 𝑧 𝑛 = π‘Ÿ 𝑛 𝑒 πœ‹ 𝑖 / 𝑛 ( 0 < π‘Ÿ 𝑛 < 1 ) such that 𝑓 ξ…ž 𝑛 ( 𝑧 𝑛 ) = 0 . This implies that 𝑓 𝑛 ( 𝑧 ) is not univalent in π‘ˆ and so the theorem is proved.

Theorem 3.2. Let (3.1) in Theorem 3.1 be satisfied. If 𝛼 β‰₯ 1 and  ( 𝛼 βˆ’ 1 ) 1 βˆ’ ( 𝐴 βˆ’ 𝐡 ) ∞  π‘š = 1 𝐡 π‘š βˆ’ 1 ξƒͺ + 𝛼 𝑛 π‘š + 1 𝑛 𝛼 2  1 βˆ’ ( 𝐴 βˆ’ 𝐡 ) ∞  π‘š = 1 𝐡 π‘š βˆ’ 1 ξƒͺ β‰₯ ( 𝑛 π‘š + 1 ) ( 𝛼 𝑛 π‘š + 1 ) 𝐴 βˆ’ 1 , 1 βˆ’ 𝐡 ( 3 . 4 ) then 𝑇 𝑛 ( 𝐴 , 𝐡 , 𝛼 ) βŠ‚ 𝑆 βˆ— 𝑛 .

Proof. We first show that ∞  π‘š = 1 𝐡 π‘š βˆ’ 1 β‰₯ 𝛼 𝑛 π‘š + 1 ∞  π‘š = 1 𝐡 π‘š βˆ’ 1 ( ( 𝑛 π‘š + 1 ) ( 𝛼 𝑛 π‘š + 1 ) 𝛼 β‰₯ 1 ) . ( 3 . 5 ) Equation (3.5) is obvious when 𝐡 β‰₯ 0 . For 0 > 𝐡 β‰₯ βˆ’ 1 , we have ∞  π‘š = 1 𝐡 π‘š βˆ’ 1 βˆ’ 𝛼 𝑛 π‘š + 1 ∞  π‘š = 1 𝐡 π‘š βˆ’ 1 ( 𝑛 π‘š + 1 ) ( 𝛼 𝑛 π‘š + 1 ) = πœ‡ 1 βˆ’ πœ‡ 2 + πœ‡ 3 βˆ’ πœ‡ 4 + β‹― + ( βˆ’ 1 ) π‘š βˆ’ 1 πœ‡ π‘š + β‹― , ( 3 . 6 ) where πœ‡ π‘š = | | 𝐡 | | 𝑛 π‘š π‘š βˆ’ 1 ( 𝑛 π‘š + 1 ) ( 𝛼 𝑛 π‘š + 1 ) > 0 . ( 3 . 7 ) Since | 𝐡 | ≀ 1 and 𝑑 ξ‚΅ π‘₯ 𝑑 π‘₯ ξ‚Ά = ( π‘₯ + 1 ) ( 𝛼 π‘₯ + 1 ) 1 βˆ’ 𝛼 π‘₯ 2 ( π‘₯ + 1 ) 2 ( 𝛼 π‘₯ + 1 ) 2 ≀ 0 ( π‘₯ β‰₯ 1 ; 𝛼 β‰₯ 1 ) , ( 3 . 8 ) { πœ‡ π‘š } is a monotonically decreasing sequence. Therefore, the inequality (3.5) follows from (3.6).
Let 𝑓 ( 𝑧 ) ∈ 𝑇 𝑛 ( 𝐴 , 𝐡 , 𝛼 ) . Then, ξ€½ 𝑓 R e ξ…ž ( 𝑧 ) + 𝛼 𝑧 𝑓 ξ…ž ξ…ž ξ€Ύ > ( 𝑧 ) 1 βˆ’ 𝐴 1 βˆ’ 𝐡 ( 𝑧 ∈ π‘ˆ ) . ( 3 . 9 ) Define 𝑝 ( 𝑧 ) in π‘ˆ by 𝑝 ( 𝑧 ) = 𝑧 𝑓 ξ…ž ( 𝑧 ) . 𝑓 ( 𝑧 ) ( 3 . 1 0 ) In view of (3.1) in Theorem 3.1 is satisfied, the function 𝑓 ( 𝑧 ) is univalent in π‘ˆ , and so 𝑝 ( 𝑧 ) = 1 + 𝑝 𝑛 𝑧 𝑛 + 𝑝 𝑛 + 1 𝑧 𝑛 + 1 + β‹― is analytic in π‘ˆ . Also it follows from (3.10) that 𝑓 ξ…ž ( 𝑧 ) + 𝛼 𝑧 𝑓 ξ…ž ξ…ž ( 𝑧 ) = ( 1 βˆ’ 𝛼 ) 𝑓 ξ…ž ( 𝑧 ) + 𝛼 𝑓 ( 𝑧 ) 𝑧 ξ€Ί 𝑧 𝑝 ξ…ž ( 𝑧 ) + ( 𝑝 ( 𝑧 ) ) 2 ξ€» . ( 3 . 1 1 )

We want to prove now that R e 𝑝 ( 𝑧 ) > 0 for 𝑧 ∈ π‘ˆ . Suppose that there exists a point 𝑧 0 ∈ π‘ˆ such that ξ€· | | 𝑧 R e 𝑝 ( 𝑧 ) > 0 | 𝑧 | < 0 | | ξ€Έ ξ€· 𝑧 , R e 𝑝 0 ξ€Έ = 0 . ( 3 . 1 2 ) Then, applying a result of Miller and Mocanu [13, Theorem 4], we have 𝑧 0 𝑝 ξ…ž ξ€· 𝑧 0 ξ€Έ + ξ€· 𝑝 ξ€· 𝑧 0 ξ€Έ ξ€Έ 2 𝑛 ≀ βˆ’ 2 ξ€· ξ€· 𝑧 R e 1 βˆ’ 𝑝 0 βˆ’ ξ€· ξ€· 𝑧 ξ€Έ ξ€Έ I m 𝑝 0 ξ€Έ ξ€Έ 2 𝑛 ≀ βˆ’ 2 . ( 3 . 1 3 ) For 𝛼 β‰₯ 1 , we deduce from Corollaries 2.2 and 2.4, (3.1), (3.5), (3.11), (3.13), and (3.4) that ξ€½ 𝑓 R e ξ…ž ξ€· 𝑧 0 ξ€Έ + 𝛼 𝑧 0 𝑓 ξ…ž ξ…ž ξ€· 𝑧 0  ξ€Έ ξ€Ύ ≀ ( 1 βˆ’ 𝛼 ) 1 βˆ’ ( 𝐴 βˆ’ 𝐡 ) ∞  π‘š = 1 𝐡 π‘š βˆ’ 1 ξƒͺ βˆ’ 𝛼 𝑛 π‘š + 1 𝑛 𝛼 2  1 βˆ’ ( 𝐴 βˆ’ 𝐡 ) ∞  π‘š = 1 𝐡 π‘š βˆ’ 1 ξƒͺ ≀ ( 𝑛 π‘š + 1 ) ( 𝛼 𝑛 π‘š + 1 ) 1 βˆ’ 𝐴 . 1 βˆ’ 𝐡 ( 3 . 1 4 ) But this contradicts (3.9) at 𝑧 = 𝑧 0 . Therefore, we must have R e 𝑝 ( 𝑧 ) > 0 ( 𝑧 ∈ π‘ˆ ) and the proof of Theorem 3.2 is completed.

Remark 3.3. In [6, Theorem 4(ii)], the authors gave the following: if 0 < 𝛼 < 1 and 𝛽 1 is the solution of the equation 1 βˆ’ 3 𝛼 2 = 𝛽 + ( 1 βˆ’ 𝛽 ) ∞  π‘š = 2 ( βˆ’ 1 ) π‘š βˆ’ 1 𝛼 + 2 ( 𝛼 βˆ’ 1 ) π‘š , π‘š ( 𝛼 ( π‘š βˆ’ 1 ) + 1 ) ( 3 . 1 5 ) then 𝑅 ( 𝛽 , 𝛼 ) βŠ‚ 𝑆 βˆ— 1 for 𝛽 β‰₯ 𝛽 1 . However, this result is not true because the series in (3.15) diverges.

4. The Radius of Convexity

Theorem 4.1. Let 𝑓 ( 𝑧 ) belong to the class 𝑇 𝑛 ( 𝛾 ) defined by 𝑇 𝑛 ( 𝛾 ) = 𝑇 𝑛 ξ‚» ( 1 , βˆ’ 1 , 𝛾 , 0 ) = 𝑓 ( 𝑧 ) ∈ 𝐴 𝑛 ∢ 𝑓 ξ…ž ξ‚€ ( 𝑧 ) β‰Ί 1 + 𝑧  1 βˆ’ 𝑧 𝛾 ξ‚Ό , ( 𝑧 ∈ π‘ˆ ) , ( 4 . 1 ) 0 < 𝛿 ≀ 1 and 0 ≀ 𝜌 < 1 . Then, ξ‚» ξ€· 𝑓 R e ( 1 βˆ’ 𝛿 ) ξ…ž ξ€Έ ( 𝑧 ) 1 / 𝛾 ξ‚΅ + 𝛿 1 + 𝑧 𝑓 ξ…ž ξ…ž ( 𝑧 ) 𝑓 ξ…ž ξ€· ( 𝑧 ) ξ‚Ά ξ‚Ό > 𝜌 | 𝑧 | < π‘Ÿ 𝑛 ξ€Έ ( 𝛾 , 𝛿 , 𝜌 ) , ( 4 . 2 ) where π‘Ÿ 𝑛 ( 𝛾 , 𝛿 , 𝜌 ) is the root in ( 0 , 1 ) of the equation ( 1 βˆ’ 2 𝛿 + 𝜌 ) π‘Ÿ 2 𝑛 βˆ’ 2 ( 1 βˆ’ 𝛿 + 𝑛 𝛿 𝛾 ) π‘Ÿ 𝑛 + 1 βˆ’ 𝜌 = 0 . ( 4 . 3 ) The result is sharp.

Proof. For 𝑓 ( 𝑧 ) ∈ 𝑇 𝑛 ( 𝛾 ) , we can write ξ€· 𝑓 ξ…ž ξ€Έ ( 𝑧 ) 1 / 𝛾 = 1 + 𝑧 𝑛 πœ‘ ( 𝑧 ) 1 βˆ’ 𝑧 𝑛 πœ‘ , ( 𝑧 ) ( 4 . 4 ) where πœ‘ ( 𝑧 ) is analytic and | πœ‘ ( 𝑧 ) | ≀ 1 in π‘ˆ . Differentiating both sides of (4.4) logarithmically, we arrive at 1 + 𝑧 𝑓 ξ…ž ξ…ž ( 𝑧 ) 𝑓 ξ…ž ( 𝑧 ) = 1 + 2 𝑛 𝛾 𝑧 𝑛 πœ‘ ( 𝑧 ) 1 βˆ’ ( 𝑧 𝑛 πœ‘ ( 𝑧 ) ) 2 + 2 𝛾 𝑧 𝑛 + 1 πœ‘ ξ…ž ( 𝑧 ) 1 βˆ’ ( 𝑧 𝑛 πœ‘ ( 𝑧 ) ) 2 ( 𝑧 ∈ π‘ˆ ) . ( 4 . 5 ) Put | 𝑧 | = π‘Ÿ < 1 and ( 𝑓 β€² ( 𝑧 ) ) 1 / 𝛾 = 𝑒 + 𝑖 𝑣 ( 𝑒 , 𝑣 ∈ 𝑅 ) . Then, (4.4) implies that 𝑧 𝑛 πœ‘ ( 𝑧 ) = 𝑒 βˆ’ 1 + 𝑖 𝑣 , 𝑒 + 1 + 𝑖 𝑣 ( 4 . 6 ) 1 βˆ’ π‘Ÿ 𝑛 1 + π‘Ÿ 𝑛 ≀ 𝑒 ≀ 1 + π‘Ÿ 𝑛 1 βˆ’ π‘Ÿ 𝑛 . ( 4 . 7 ) With the help of the Carathéodory inequality | | πœ‘ ξ…ž | | ≀ | | | | ( 𝑧 ) 1 βˆ’ πœ‘ ( 𝑧 ) 2 1 βˆ’ π‘Ÿ 2 , ( 4 . 8 ) it follows from (4.5) and (4.6) that ξ‚» ξ€· 𝑓 R e ( 1 βˆ’ 𝛿 ) ξ…ž ξ€Έ ( 𝑧 ) 1 / 𝛾 ξ‚΅ + 𝛿 1 + 𝑧 𝑓 ξ…ž ξ…ž ( 𝑧 ) 𝑓 ξ…ž ξ‚» 𝑧 ( 𝑧 ) ξ‚Ά ξ‚Ό β‰₯ ( 1 βˆ’ 𝛿 ) 𝑒 + 𝛿 + 2 𝑛 𝛿 𝛾 R e 𝑛 πœ‘ ( 𝑧 ) 1 βˆ’ ( 𝑧 𝑛 πœ‘ ( 𝑧 ) ) 2 ξ‚Ό | | | | 𝑧 βˆ’ 2 𝛿 𝛾 𝑛 + 1 πœ‘ ξ…ž ( 𝑧 ) 1 βˆ’ ( 𝑧 𝑛 πœ‘ ( 𝑧 ) ) 2 | | | | β‰₯ ( 1 βˆ’ 𝛿 ) 𝑒 + 𝛿 + 𝑛 𝛿 𝛾 2 ξ‚€ 𝑒 𝑒 βˆ’ 𝑒 2 + 𝑣 2  + 𝛿 𝛾 2 ( 𝑒 βˆ’ 1 ) 2 + 𝑣 2 βˆ’ π‘Ÿ 2 𝑛 ξ€· ( 𝑒 + 1 ) 2 + 𝑣 2 ξ€Έ π‘Ÿ 𝑛 βˆ’ 1 ξ€· 1 βˆ’ π‘Ÿ 2 𝑒 ξ€Έ ξ€· 2 + 𝑣 2 ξ€Έ 1 / 2 = 𝐹 𝑛 ( 𝑒 , 𝑣 ) ( s a y πœ• ) , ( 4 . 9 ) 𝐹 πœ• 𝑣 𝑛 ( 𝑒 , 𝑣 ) = 𝛿 𝛾 𝑣 𝐺 𝑛 ( 𝑒 , 𝑣 ) , ( 4 . 1 0 ) where 0 < π‘Ÿ < 1 , 0 < 𝛿 ≀ 1 and 𝐺 𝑛 ( 𝑒 , 𝑣 ) = 𝑛 𝑒 ξ€· 𝑒 2 + 𝑣 2 ξ€Έ 2 + 1 βˆ’ π‘Ÿ 2 𝑛 π‘Ÿ 𝑛 βˆ’ 1 ξ€· 1 βˆ’ π‘Ÿ 2 𝑒 ξ€Έ ξ€· 2 + 𝑣 2 ξ€Έ 1 / 2 + π‘Ÿ 2 𝑛 ξ€· ( 𝑒 + 1 ) 2 + 𝑣 2 ξ€Έ βˆ’ ξ€· ( 𝑒 βˆ’ 1 ) 2 + 𝑣 2 ξ€Έ 2 π‘Ÿ 𝑛 βˆ’ 1 ξ€· 1 βˆ’ π‘Ÿ 2 𝑒 ξ€Έ ξ€· 2 + 𝑣 2 ξ€Έ 3 / 2 > 0 ( 4 . 1 1 ) because of (4.6) and (4.7). In view of (4.10) and (4.11), we see that 𝐹 𝑛 ( 𝑒 , 𝑣 ) β‰₯ 𝐹 𝑛 ( 𝑒 , 0 ) = ( 1 βˆ’ 𝛿 ) 𝑒 + 𝛿 + 𝑛 𝛿 𝛾 2 ξ‚€ 1 𝑒 βˆ’ 𝑒  + 𝛿 𝛾 2 π‘Ÿ 𝑛 βˆ’ 1 ξ€· 1 βˆ’ π‘Ÿ 2 ξ€Έ Γ—  ξ€· 1 βˆ’ π‘Ÿ 2 𝑛 ξ€Έ ξ‚€ 1 𝑒 + 𝑒  ξ€· βˆ’ 2 1 + π‘Ÿ 2 𝑛 ξ€Έ  . ( 4 . 1 2 )

Let us now calculate the minimum value of 𝐹 𝑛 ( 𝑒 , 0 ) on the closed interval [ ( 1 βˆ’ π‘Ÿ 𝑛 ) / ( 1 + π‘Ÿ 𝑛 ) , ( 1 + π‘Ÿ 𝑛 ) / ( 1 βˆ’ π‘Ÿ 𝑛 ) ] . Noting that 1 βˆ’ π‘Ÿ 2 𝑛 π‘Ÿ 𝑛 βˆ’ 1 ξ€· 1 βˆ’ π‘Ÿ 2 ξ€Έ [ 8 ] ) β‰₯ 𝑛 ( s e e ( 4 . 1 3 ) and (4.7), we deduce from (4.12) that 𝑑 𝐹 𝑑 𝑒 𝑛 ( 𝑒 , 0 ) = 1 βˆ’ 𝛿 + 𝛿 𝛾 2   1 βˆ’ π‘Ÿ 2 𝑛 π‘Ÿ 𝑛 βˆ’ 1 ξ€· 1 βˆ’ π‘Ÿ 2 ξ€Έ ξƒͺ βˆ’ 1 + 𝑛 𝑒 2  1 βˆ’ π‘Ÿ 2 𝑛 π‘Ÿ 𝑛 βˆ’ 1 ξ€· 1 βˆ’ π‘Ÿ 2 ξ€Έ βˆ’ 𝑛 ξƒͺ ξƒ­ β‰₯ 1 βˆ’ 𝛿 + 𝛿 𝛾 2   1 βˆ’ π‘Ÿ 2 𝑛 π‘Ÿ 𝑛 βˆ’ 1 ξ€· 1 βˆ’ π‘Ÿ 2 ξ€Έ ξƒͺ βˆ’ ξ‚΅ + 𝑛 1 + π‘Ÿ 𝑛 1 βˆ’ π‘Ÿ 𝑛 ξ‚Ά 2  1 βˆ’ π‘Ÿ 2 𝑛 π‘Ÿ 𝑛 βˆ’ 1 ξ€· 1 βˆ’ π‘Ÿ 2 ξ€Έ βˆ’ 𝑛 ξƒͺ ξƒ­ = 1 βˆ’ 𝛿 + 2 𝛿 𝛾 𝐼 𝑛 ( π‘Ÿ ) ( 1 βˆ’ π‘Ÿ 𝑛 ) 2 , ( 4 . 1 4 ) where 𝐼 𝑛 𝑛 ( π‘Ÿ ) = 2 ξ€· 1 + π‘Ÿ 2 𝑛 ξ€Έ ξ€· βˆ’ π‘Ÿ 1 + π‘Ÿ 2 + β‹― + π‘Ÿ 2 𝑛 βˆ’ 2 ξ€Έ . ( 4 . 1 5 ) Also 𝐼 ξ…ž 𝑛 ( π‘Ÿ ) = 𝑛 2 π‘Ÿ 2 𝑛 βˆ’ 1 βˆ’ ξ€· 1 + 3 π‘Ÿ 2 + β‹― + ( 2 𝑛 βˆ’ 1 ) π‘Ÿ 2 𝑛 βˆ’ 2 ξ€Έ ( 4 . 1 6 ) and 𝐼 ξ…ž 1 ( π‘Ÿ ) = π‘Ÿ βˆ’ 1 < 0 . Suppose that 𝐼 ξ…ž 𝑛 ( π‘Ÿ ) < 0 . Then, 𝐼 ξ…ž 𝑛 + 1 ( π‘Ÿ ) = ( 𝑛 + 1 ) 2 π‘Ÿ 2 𝑛 + 1 βˆ’ ( 2 𝑛 + 1 ) π‘Ÿ 2 𝑛 βˆ’ ξ€· 1 + 3 π‘Ÿ 2 + β‹― + ( 2 𝑛 βˆ’ 1 ) π‘Ÿ 2 𝑛 βˆ’ 2 ξ€Έ < 𝑛 2 π‘Ÿ 2 𝑛 βˆ’ ξ€· 1 + 3 π‘Ÿ 2 + β‹― + ( 2 𝑛 βˆ’ 1 ) π‘Ÿ 2 𝑛 βˆ’ 2 ξ€Έ < 𝐼 ξ…ž 𝑛 ( π‘Ÿ ) < 0 . ( 4 . 1 7 ) Hence, by virtue of the mathematical induction, we have 𝐼 ξ…ž 𝑛 ( π‘Ÿ ) < 0 for all 𝑛 ∈ 𝑁 and 0 ≀ π‘Ÿ < 1 . This implies that 𝐼 𝑛 ( π‘Ÿ ) > 𝐼 𝑛 ( 1 ) = 0 ( 𝑛 ∈ 𝑁 ; 0 ≀ π‘Ÿ < 1 ) . ( 4 . 1 8 ) In view of (4.14) and (4.18), we see that 𝑑 𝐹 𝑑 𝑒 𝑛 ξ‚΅ ( 𝑒 , 0 ) > 0 1 βˆ’ π‘Ÿ 𝑛 1 + π‘Ÿ 𝑛 ≀ 𝑒 ≀ 1 + π‘Ÿ 𝑛 1 βˆ’ π‘Ÿ 𝑛 ξ‚Ά . ( 4 . 1 9 ) Further it follows from (4.9), (4.12), and (4.19) that ξ‚» ξ€· 𝑓 R e ( 1 βˆ’ 𝛿 ) ξ…ž ξ€Έ ( 𝑧 ) 1 / 𝛾 ξ‚΅ + 𝛿 1 + 𝑧 𝑓 ξ…ž ξ…ž ( 𝑧 ) 𝑓 ξ…ž ( 𝑧 ) ξ‚Ά ξ‚Ό βˆ’ 𝜌 β‰₯ 𝐹 𝑛 ξ‚΅ 1 βˆ’ π‘Ÿ 𝑛 1 + π‘Ÿ 𝑛 ξ‚Ά , 0 βˆ’ 𝜌 = ( 1 βˆ’ 𝛿 ) 1 βˆ’ π‘Ÿ 𝑛 1 + π‘Ÿ 𝑛 + 𝛿 1 βˆ’ 2 𝑛 𝛾 π‘Ÿ 𝑛 βˆ’ π‘Ÿ 2 𝑛 1 βˆ’ π‘Ÿ 2 𝑛 = 𝐽 βˆ’ 𝜌 𝑛 ( π‘Ÿ ) 1 βˆ’ π‘Ÿ 2 𝑛 , ( 4 . 2 0 ) where 0 ≀ 𝜌 < 1 and 𝐽 𝑛 ( π‘Ÿ ) = ( 1 βˆ’ 2 𝛿 + 𝜌 ) π‘Ÿ 2 𝑛 βˆ’ 2 ( 1 βˆ’ 𝛿 + 𝑛 𝛿 𝛾 ) π‘Ÿ 𝑛 + 1 βˆ’ 𝜌 . ( 4 . 2 1 ) Note that 𝐽 𝑛 ( 0 ) = 1 βˆ’ 𝜌 > 0 and 𝐽 𝑛 ( 1 ) = βˆ’ 2 𝑛 𝛿 𝛾 < 0 . If we let π‘Ÿ 𝑛 ( 𝛾 , 𝛿 , 𝜌 ) denote the root in ( 0 , 1 ) of the equation 𝐽 𝑛 ( π‘Ÿ ) = 0 , then (4.20) yields the desired result (4.2).

To see that the bound π‘Ÿ 𝑛 ( 𝛾 , 𝛿 , 𝜌 ) is the best possible, we consider the function ξ€œ 𝑓 ( 𝑧 ) = 𝑧 0 ξ‚΅ 1 βˆ’ 𝑑 𝑛 1 + 𝑑 𝑛 ξ‚Ά 𝛾 𝑑 𝑑 ∈ 𝑇 𝑛 ( 𝛾 ) . ( 4 . 2 2 ) It is clear that for 𝑧 = π‘Ÿ ∈ ( π‘Ÿ 𝑛 ( 𝛾 , 𝛿 , 𝜌 ) , 1 ) , ξ€· 𝑓 ( 1 βˆ’ 𝛿 ) ξ…ž ξ€Έ ( π‘Ÿ ) 1 / 𝛾 ξ‚΅ + 𝛿 1 + π‘Ÿ 𝑓 ξ…ž ξ…ž ( π‘Ÿ ) 𝑓 ξ…ž ξ‚Ά 𝐽 ( π‘Ÿ ) βˆ’ 𝜌 = 𝑛 ( π‘Ÿ ) 1 βˆ’ π‘Ÿ 2 𝑛 < 0 , ( 4 . 2 3 ) which shows that the bound π‘Ÿ 𝑛 ( 𝛾 , 𝛿 , 𝜌 ) cannot be increased.

Setting 𝛿 = 1 , Theorem 4.1 reduces to the following result.

Corollary 4.2. Let 𝑓 ( 𝑧 ) ∈ 𝑇 𝑛 ( 𝛾 ) and 0 ≀ 𝜌 < 1 . Then, 𝑓 ( 𝑧 ) is convex of order 𝜌 in  ξ€· ( | 𝑧 | < 𝑛 𝛾 ) 2 + ( 1 βˆ’ 𝜌 ) 2 ξ€Έ 1 / 2 βˆ’ 𝑛 𝛾 ξƒ­ 1 βˆ’ 𝜌 1 / 𝑛 . ( 4 . 2 4 ) The result is sharp.

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