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Integral transforms of a class of analytic functions. (English) Zbl 1146.30009

Let \(\Delta\) be the complex unit disc and \(\mathcal A\) be the class of all analytic functions in \(\Delta\), normalized with the conditions \(f(0)=f'(0)-1=0\). A function in \(\mathcal A\) is said to be in the class \(\mathcal P_{\lambda}(\beta)\) if \(\mathrm{Re}[\mathrm{e}^{\mathrm{i}\phi}(f'(z)+\gamma zf''(z)-\beta]>0\) in \(\Delta\) (\(\phi\in\mathbb{R}\), \(\gamma\geq 0\) and \(\beta<1\)). For a nonnegative real-valued integrable function \(\lambda(t)\) satisfying the normalizing condition \(\int_0^1\lambda(t)dt=1\) and \(f\in\mathcal A\) let
\[ F(z)=V_{\lambda}(f)(z)=\int_0^1\lambda(t)\frac{f(tz)}{t}dt \] and
\[ \Lambda_{\gamma}(t)=\int_t^1\frac{\lambda(s)}{s^{1/\gamma}} ds,\;\;\gamma>0 \]
\[ \Pi_{\gamma}(t)=\int_t^1\Lambda_{\gamma}(s)s^{1/\gamma-2}ds \text{ for }gamma>0\text{ and } \Pi_{\gamma}(t)=\int_t^1\frac{\lambda(s)}{s}ds\text{ for }\gamma=0 \]
If
\[ \frac{\beta}{1-\beta}=-\int_0^1\lambda(t)g_{\gamma}(t)dt \]
for some \(\lambda\geq 0\) and \(\beta< 1\) and if, in addition \(\Pi_{\gamma}(t)/(1-t^2)\) is decreasing on (0,1), the authors prove their principal result, that states that \(V_{\lambda}(\mathcal {P}_{\lambda}(\beta))\subset S^*\), where \(S^*\) is the subclass of \(\mathcal A\) consisting of starlike functions in \(\Delta\). Some other results on the class \(\mathcal P_{\gamma}(\beta)\) and applications are also given.

MSC:

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

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