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 [1–5]). 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 CarathΓ©odory 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.