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Subordinants of differential superordinations. (English) Zbl 1039.30011
Let $${\mathcal H}$$ be the class of functions analytic in $$U$$ and $${\mathcal H}(a,n)$$ be the subclass of $${\mathcal H}$$ 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 $${\mathbb C}$$, let $$p\in {\mathcal H}$$ and let $$\phi(r,s,t;z):{\mathbb C}^3\times U \rightarrow {\mathbb C}$$. In the present paper, the authors obtain conditions on $$\Omega$$, $$\Delta$$ and $$\phi$$ for which the following implication holds: $$\Omega\subset \{\phi(p(z),zp'(z),z^2p''(z);z)| z\in U\}\Rightarrow \Delta\subset p(U)$$.
When $$\Omega$$ and $$\Delta$$ are simply connected domains with $$\Omega,\Delta\not={\mathbb C}$$, 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 $${\mathcal Q}(a)$$, 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 $${\mathbb C}$$ and $$q\in{\mathcal H}(a,n)$$ with $$q'(z) \not=0$$, the class of admissible functions $$\Phi_n [\Omega,q]$$ consists of those functions $$\phi:{\mathbb C}^3\times \overline {U} \rightarrow {\mathbb C}$$ 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 $$\phi:{\mathbb C}^3\times U\rightarrow {\mathbb C}$$. Suppose that $$\phi(q(z),zq'(z),z^2q''(z);z)=h(z)$$ has a solution $$q\in {\mathcal Q}(a)$$. If $$\phi\in\Phi_n[h(U),q]$$, $$p\in{\mathcal Q}(a)$$ 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 T. Bulboaca [Demonstr. Math. 35, No. 2, 287–292 (2002; Zbl 1010.30020)].

##### MSC:
 30C80 Maximum principle, Schwarz’s lemma, Lindelöf principle, analogues and generalizations; subordination 30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.) 34A40 Differential inequalities involving functions of a single real variable 30C40 Kernel functions in one complex variable and applications
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