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Convolution properties of some classes of analytic functions. (English) Zbl 0886.30019

Kuz’mina, G. V. (ed.) et al., Analytical number theory and function theory. 13. Work collection. Dedicated to the 80th birthday of Yu. V. Linnik. Sankt-Peterburg: Nauka. Zap. Nauchn. Semin. POMI. 226, 138-154 (1996).
Let \({\mathcal A}\) denote the class of functions analytic in the unit disc \(\Delta= \{z:|z|<1\}\) normalized by conditions \(f(0)=0\), \(f'(0)=1\) and let \(R(\alpha,\beta) \subset {\mathcal A}\) be the class of functions \(f\in{\mathcal A}\) such that \[ \text{Re}\bigl[f'(z) +\alpha zf''(z) \bigr]> \beta\text{ for } z\in\Delta, \] where \(\text{Re} \alpha>0\), \(\beta<1\). For two functions \(f(z)= \sum^\infty_{k=0} a_kz^k\), \(g(z)= \sum^\infty_{k=0} b_kz^k\) belonging to \({\mathcal A}\) we define convolution \(f*g\) as follows \[ (f*g)(z)= \sum^\infty_{k=0} a_kb_kz^k. \] In this paper many problems concerning convolution “*” and subordination “\(\prec\)” are studied. Among others it is proved: I. For \(\alpha_1\geq \alpha_2>0\) let \(R(\alpha_1, \beta_1) \subset S^*\) and \(R (\alpha_1, \beta_2) \subset S^*\) where \(S^*\) denote the class of starlike functions. If \(f\in R(\alpha_1, \beta)\), \(g\in R(\alpha_2\beta)\) where \(\beta= \max\{\beta_1, \beta_2\}\) then \(h=f* g\in K\) where \(K\) denotes the class of convex functions, provided \(1-\beta_2 \geq 4(1-\beta)^2 \left(1-\int^1_0 {dt\over 1+t^{\alpha_1}} \right)\). II. Let \(\mu>0\), \(0<\lambda_1< \lambda<1\), \(\lambda_1= \lambda \mu/(1+\mu)\), \(\rho= (\lambda+ \lambda_1)/(1- \lambda_2)\) and \(\beta= \begin{cases} {1- \lambda \over 1-\lambda_1}, \quad & 0<\lambda +\lambda_1\leq 1,\\ {1-\lambda^2_1-\lambda^2 \over 2(1-\lambda^2_1)}, \quad & \lambda^2_1+ \lambda^2 \leq 1<\lambda +\lambda_1 \end{cases}\). If \(f\in {\mathcal A}\) satisfies the condition \(f'(z) \left({f(z) \over z} \right)^{\mu-1} \prec 1+ \lambda z\); then \(\text{Re} {zf'(z) \over f(z)} >\beta\), for \(z\in\Delta\), and \(|{zf'(z) \over f(z)}-1 |\leq\rho\), for \(z\in\Delta\).
For the entire collection see [Zbl 0868.00016].

MSC:

30C75 Extremal problems for conformal and quasiconformal mappings, other methods
30C55 General theory of univalent and multivalent functions of one complex variable
30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
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