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Subordinants of differential superordinations. (English) Zbl 1039.30011
Let ${\cal H}$ be the class of functions analytic in $U$ and ${\cal H}(a,n)$ be the subclass of ${\cal H}$ consisting of functions of the form $f(z)=a+a_nz^n+a_{n+1}z^{n+1}+\ldots$. Let $\Omega$ and $\Delta$ be any sets in the complex plane ${\Bbb C}$, let $p\in {\cal H}$ and let $\phi(r,s,t;z):{\Bbb C}^3\times U \rightarrow {\Bbb C}$. In the present paper, the authors obtain conditions on $\Omega$, $\Delta$ and $\phi$ for which the following implication holds: $\Omega\subset \{\phi(p(z),zp'(z),z^2p''(z);z)\vert z\in U\}\Rightarrow \Delta\subset p(U)$. When $\Omega$ and $\Delta$ are simply connected domains with $\Omega,\Delta\not={\Bbb C}$, the above implication becomes $ h(z)\prec \phi(p(z),zp'(z),z^2p''(z);z) \Rightarrow q(z)\prec p(z)$, where $h$ and $q$ are the conformal mappings of $U$ onto the domains $\Omega$ and $\Delta$ respectively. If $p$ and $\phi(p(z),zp'(z),z^2p''(z);z)$ are univalent and if $p$ satisfies the second order superordination $h(z)\prec \phi(p(z),zp'(z),z^2p''(z);z)$, $p$ is the solution of the differential superordination. (If $f$ is subordinate to $F$, then $F$ is superordinate to $f$.) Subordinant and best subordinant are defined similarly like dominant and best dominant in case of differential subordination. Denote by ${\cal Q}(a)$, the set of all functions $f(z)$, with $f(0)=a$, that are analytic and injective on $\overline{U}-E(f)$, where $E(f)=\{\zeta \in\partial U: \lim_{z\rightarrow \zeta} f(z)=\infty \}$, and are such that $f'(\zeta)\not=0$ for $\zeta\in\partial U-E(f)$. For a set $\Omega$ in ${\Bbb C}$ and $q\in{\cal H}(a,n)$ with $q'(z) \not=0$, the class of admissible functions $\Phi_n [\Omega,q]$ consists of those functions $\phi:{\Bbb C}^3\times \overline {U} \rightarrow {\Bbb C}$ that satisfy the admissibility condition: $\phi(r,s,t;\zeta)\in\Omega$, whenever $r=q(z)$, $s=zq'(z)/m$, $\text{Re}(t/s)+1\leq (1/m)\text{Re} [zq''(z)/q'(z) +1]$, where $\zeta\in\partial U$, $z\in U$ and $m\geq n\geq 1$. The principal result proved in the paper for second order differential superordinations is the following: Theorem. Let $h$ be analytic in $U$ and $\phi:{\Bbb C}^3\times U\rightarrow {\Bbb C}$. Suppose that $\phi(q(z),zq'(z),z^2q''(z);z)=h(z)$ has a solution $q\in {\cal Q}(a)$. If $\phi\in\Phi_n[h(U),q]$, $p\in{\cal Q}(a)$ and $\phi(p(z),zp'(z),z^2p''(z);z)$ is univalent in $U$, then $h(z)\prec \phi(p(z),zp'(z),z^2p''(z);z) \Rightarrow q\prec p$ and $q$ is the best subordinant. By using the results for first order superordinations together with previously known results for differential subordinations, the authors have obtained several differential “sandwich theorems”. Also a special second order differential superordination is considered. Some applications of the results of this paper was obtained recently by {\it T. Bulboaca} [Demonstr. Math. 35, No. 2, 287--292 (2002; Zbl 1010.30020)].

30C80Maximum principle; Schwarz’s lemma, Lindelöf principle, etc. (one complex variable)
30C45Special classes of univalent and multivalent functions
34A40Differential inequalities (ODE)
30C40Kernel functions and applications (one complex variable)
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