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Sandwich-type theorems for a class of integral operators. (English) Zbl 1154.30020
Let $H(U)$ be the space of analytic functions of the unit disk $U$. By $f \prec g$ subordination is denoted, i.e. $f = g \circ \omega$ with $\vert \omega\vert < 1$ and $\omega(0) = 0.$ For a given function $h \in H(U)$ the authors define the integral operator $I_{h;\beta} : K \rightarrow H(U),$ with $K \subset H(U),$ by $$ I_{h;\beta}[f](z) = \left[\beta \int^z_0 f^{\beta} (t)h^{-1}(t)h'(t) \, dt \right]^{1/\beta} ,$$ where $\beta \in {\Bbb C}$ and all powers are the principal ones. The authors determine sufficient conditions on $g_1, g_2$ and $\beta$ such that $$ \left[zh'(z) \over h(z) \right]^{1/ \beta} g_1(z) \prec \left[zh'(z) \over h(z) \right]^{1/ \beta} f(z) \prec \left[zh'(z) \over h(z) \right]^{1/ \beta} g_2(z)$$ implies $$I_{h; \beta}[g_1](z) \prec I_{h; \beta}[f](z) \prec I_{h; \beta} [g_2] (z).$$

30C80Maximum principle; Schwarz’s lemma, Lindelöf principle, etc. (one complex variable)
30C45Special classes of univalent and multivalent functions
45P05Integral operators
Full Text: Euclid