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Some operational techniques in the theory of analytic functions. (English) Zbl 0622.30007

Let A be the class of functions \(f(z) = z+\sum^{\infty}_{n=1} a_{n+1}z^{n+1}\) analytic in the unit disk, and T its subclass with \(a_{n+1}\leq 0\). The authors state without proof various results on coefficient estimates and distortion theorems for such classes by making use of a certain linear operator defined in terms of Hadamard product. Let V(\(\alpha)\) with \(\alpha >-1\) be the subclass of A characterized by Re\((zf'(z)/f(z))>-\alpha.\) A linear operator \(L(a,c)\) on A with \(c\neq 0,- 1,-2,..\). is defined by \[ L(a,c)f=\Phi (a,c)*f\text{ where } \Phi (a,c;z)=z_ 2F_ 1(1,a;c;z), \] and a class \(V(a,c;\alpha)\subset A\) is characterized by \(L(a,c)f\in V(\alpha)\). Any class associated with T instead of A will be denoted by affixing the suffix zero to the corresponding one. Typical theorems on coefficient estimate and starlikeness are then stated as follows: If \(f\in A\) satisfies \[ \sum^{\infty}_{n=1}((2+\alpha)_ n/(1+\alpha)_ n)| (a)_ n/(c)_ n| | a_{n+1}| \leq 1, \] then \(f\in V(a,c;\alpha)\) where \((\lambda)_ n=\Gamma (\lambda +n)/\Gamma (\lambda)\), and this estimate is sharp: If \(f\in T\) is in the class \(V_ 0(a,c;\alpha)\) with \((a)_ n/(c)_ n>0\), then f is starlike of order \(\delta\) with \(0\leq \delta <1\) in \(\{| z| <r\}\) where \[ r=\inf ((1- \delta)(2+\alpha)_ n(a)_ n/(n+1-\delta)(1+\alpha)_ n(c)_ n)^{1/n}, \] the infimum being taken over natural numbers.
Similar theorems concerning with the subclass of A or T characterized by Re\((1+zf''(z)/f'(z))>-\alpha\) instead of Re\((zf'(z)/f(z))>-\alpha\) are derived. Finally, characterization theorems involving an integral functional and a fractional calculus operator are stated.
Reviewer: Y.Komatu

MSC:

30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
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