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Convexity of integral transforms and duality. (English) Zbl 1381.30006

Summary: For \(\lambda\) satisfying a certain admissibility criteria, sufficient conditions are obtained for the integral transform \[ v_\lambda(f)(z):=\int_0^1\lambda(t)\frac{f(tz)}{t}\mathrm{d}t \] to map normalized analytic functions \(f\) satisfying \[ \operatorname{Re}^{i\phi}\left((1-\alpha+2\gamma)\frac{fz}{z}+(\alpha-2\gamma)f'(z)+\gamma zf''(z)-\beta \right)>0 \] into the class of convex functions. Several interesting applications for different choices of \(\lambda\) are discussed. In particular, the smallest value \(\beta<1\) is obtained that ensures a function \(f\) satisfying \(\operatorname{Re}(f'(z)+\alpha zf''(z) + \gamma z^2f'''(z)) >\beta\) is convex.

MSC:

30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
30C80 Maximum principle, Schwarz’s lemma, Lindelöf principle, analogues and generalizations; subordination
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