Subordinants of differential superordinations.

*(English)*Zbl 1039.30011Let \({\mathcal H}\) 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)].

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)].

Reviewer: Vravi Ravichandran (Penang)

##### MSC:

30C80 | Maximum principle, Schwarz’s lemma, LindelĂ¶f 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 |