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On some subclass of the class of convex functions in the annulus. (English) Zbl 0558.30013

Let \(S(b_ 1)\), \(0<b_ 1\leq 1\), denote the class of all functions f of the form \(f(z)=b_ 1[z+a_ 2z^ 2+...+a_ nz^ n+...]\) univalent in the unit disc satisfying the condition \(| f(z)| <1\). In the present paper, it is shown that the non-homogeneous functional \(I[f]=Re[a_ 4-pa_ 2a_ 3+qa^ 2_ 2+ra_ 2],\) for all p,q,r\(\in {\mathbb{R}}\), \(r>0\) sufficiently close to 1 is maximized only by the function \(f^*\) defined by \(f^*(z)=p^{-1}[b_ 1p(z)],\quad p(z)=z/(1-z)^ 2.\) This result is sharp.
Reviewer: K.I.Noor

MSC:

30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
30C50 Coefficient problems for univalent and multivalent functions of one complex variable
30C75 Extremal problems for conformal and quasiconformal mappings, other methods
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