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Subordinants of differential superordinations. (English) Zbl 1039.30011

Let $ℋ$ be the class of functions analytic in $U$ and $ℋ\left(a,n\right)$ be the subclass of $ℋ$ consisting of functions of the form $f\left(z\right)=a+{a}_{n}{z}^{n}+{a}_{n+1}{z}^{n+1}+...$. Let ${\Omega }$ and ${\Delta }$ be any sets in the complex plane $ℂ$, let $p\in ℋ$ and let $\phi \left(r,s,t;z\right):{ℂ}^{3}×U\to ℂ$. In the present paper, the authors obtain conditions on ${\Omega }$, ${\Delta }$ and $\phi$ for which the following implication holds: ${\Omega }\subset \left\{\phi \left(p\left(z\right),z{p}^{\text{'}}\left(z\right),{z}^{2}{p}^{\text{'}\text{'}}\left(z\right);z\right)|z\in U\right\}⇒{\Delta }\subset p\left(U\right)$.

When ${\Omega }$ and ${\Delta }$ are simply connected domains with ${\Omega },{\Delta }\ne ℂ$, the above implication becomes $h\left(z\right)\prec \phi \left(p\left(z\right),z{p}^{\text{'}}\left(z\right),{z}^{2}{p}^{\text{'}\text{'}}\left(z\right);z\right)⇒q\left(z\right)\prec p\left(z\right)$, where $h$ and $q$ are the conformal mappings of $U$ onto the domains ${\Omega }$ and ${\Delta }$ respectively. If $p$ and $\phi \left(p\left(z\right),z{p}^{\text{'}}\left(z\right),{z}^{2}{p}^{\text{'}\text{'}}\left(z\right);z\right)$ are univalent and if $p$ satisfies the second order superordination $h\left(z\right)\prec \phi \left(p\left(z\right),z{p}^{\text{'}}\left(z\right),{z}^{2}{p}^{\text{'}\text{'}}\left(z\right);z\right)$, $p$ 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 $𝒬\left(a\right)$, the set of all functions $f\left(z\right)$, with $f\left(0\right)=a$, that are analytic and injective on $\overline{U}-E\left(f\right)$, where $E\left(f\right)=\left\{\zeta \in \partial U:{lim}_{z\to \zeta }f\left(z\right)=\infty \right\}$, and are such that ${f}^{\text{'}}\left(\zeta \right)\ne 0$ for $\zeta \in \partial U-E\left(f\right)$. For a set ${\Omega }$ in $ℂ$ and $q\in ℋ\left(a,n\right)$ with ${q}^{\text{'}}\left(z\right)\ne 0$, the class of admissible functions ${{\Phi }}_{n}\left[{\Omega },q\right]$ consists of those functions $\phi :{ℂ}^{3}×\overline{U}\to ℂ$ that satisfy the admissibility condition: $\phi \left(r,s,t;\zeta \right)\in {\Omega }$, whenever $r=q\left(z\right)$, $s=z{q}^{\text{'}}\left(z\right)/m$, $\text{Re}\left(t/s\right)+1\le \left(1/m\right)\text{Re}\left[z{q}^{\text{'}\text{'}}\left(z\right)/{q}^{\text{'}}\left(z\right)+1\right]$, where $\zeta \in \partial U$, $z\in U$ and $m\ge n\ge 1$.

The principal result proved in the paper for second order differential superordinations is the following:

Theorem. Let $h$ be analytic in $U$ and $\phi :{ℂ}^{3}×U\to ℂ$. Suppose that $\phi \left(q\left(z\right),z{q}^{\text{'}}\left(z\right),{z}^{2}{q}^{\text{'}\text{'}}\left(z\right);z\right)=h\left(z\right)$ has a solution $q\in 𝒬\left(a\right)$. If $\phi \in {{\Phi }}_{n}\left[h\left(U\right),q\right]$, $p\in 𝒬\left(a\right)$ and $\phi \left(p\left(z\right),z{p}^{\text{'}}\left(z\right),{z}^{2}{p}^{\text{'}\text{'}}\left(z\right);z\right)$ is univalent in $U$, then $h\left(z\right)\prec \phi \left(p\left(z\right),z{p}^{\text{'}}\left(z\right),{z}^{2}{p}^{\text{'}\text{'}}\left(z\right);z\right)⇒q\prec p$ 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 T. Bulboaca [Demonstr. Math. 35, No. 2, 287–292 (2002; Zbl 1010.30020)].

##### MSC:
 30C80 Maximum principle; Schwarz’s lemma, Lindelöf principle, etc. (one complex variable) 30C45 Special classes of univalent and multivalent functions 34A40 Differential inequalities (ODE) 30C40 Kernel functions and applications (one complex variable)