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On the range of values of analytic functions relating to a family of integral operators. (English) Zbl 0659.30010

The author carries forward his studies [Ann. Pol. Math. 46, 141-145 (1985; Zbl 0604.30013)]. Let \({\mathcal J}(\alpha)\) denote the class of functions f analytic in the unit disc D(0,1) with \[ f(0)=f'(0)-1=0,\quad {\mathcal R}(f(z)/z)>\alpha \quad (z\in D(0,1)), \] for some \(\alpha\) less than 1. For a continuous parameter \(\lambda\) (\(\geq 0)\) a family of operators \[ {\mathcal L}^{\lambda}f(z)=\int_{I}f(zt)/t d\sigma_{\lambda}(t) \] is defined on \({\mathcal J}(\alpha)\) for probability measures \(\sigma_{\lambda}(t)\) on \([0,1]=I\). When \(a>0\) let \[ \sigma_{\lambda}(t,a)=a^{\lambda}/\Gamma (\lambda)\int^{t}_{0}\tau^{a-1}(\log 1/\tau)^{\lambda -1} d\tau, \] and define an operator \((a)^{\alpha}\) by \[ (a)^{\lambda}f(z)=a^{\lambda}/\Gamma (\lambda)\int_{I}f(zt)t^{a- 2}(\log 1/t)^{\lambda -1} dt. \] When \(| z| \leq r<1\) the distortion inequality \[ | \frac{{\mathcal L}(a)^{\lambda}f(z)}{z}- \frac{{\mathcal L}(a)^{\lambda}\phi (r,a)}{r}| \leq \frac{{\mathcal L}(a)^{\lambda}\psi (r,a)}{r}-1 \] is obtained, where \[ \frac{{\mathcal L}(a)^{\lambda}\phi (r,a)}{r}=1+2(1- \alpha)\sum^{\infty}_{n=1}(\frac{a}{2n+a})^{\lambda} r^{2n}, \]
\[ \frac{{\mathcal L}(a)^{\lambda}\psi (r,a)}{r}=1+2(1- \alpha)\sum^{\infty}_{n=1}(\frac{a}{2n+a-1})^{\lambda} r^{2n-1}, \] and extremal functions are noted. The range of values of \({\mathcal L}(a)^{\lambda}f(z)/z\) is estimated with particular reference to the cases \(\lambda >1\) and \(0<\lambda \leq 1\).
Reviewer: C.N.Linden

MSC:

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

Citations:

Zbl 0604.30013
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