## Rocky Mountain Journal of Mathematics

- Rocky Mountain J. Math.
- Volume 47, Number 2 (2017), 383-390.

### Application of strong differential superordination to a general equation

R. Aghalary, P. Arjomandinia, and A. Ebadian

#### Abstract

In this paper, we study the notion of strong differential superordination as a dual concept of strong differential subordination, introduced in~\cite {1.a}. The notion of strong differential superordination has recently been studied by many authors, see, for example, \cite {2.a, 3.a, 5.a}. Let $q(z)$ be an analytic function in $\mathbb {D}$ that satisfies the first order differential equation $$\theta (q(z))+F(z)q'(z)\varphi (q(z))=h(z).$$ \smallskip Suppose that $p(z)$ is analytic and univalent in the closure of the open unit disk $\overline {\mathbb {D}}$ with $p(0)=q(0)$. We shall find conditions on $h(z),G(z),\theta (z)$ and $\varphi (z)$ such that $$ h(z)\prec \prec \theta (p(z))+\frac {G(\xi )}{\xi }zp'(z)\varphi (p(z))\Longrightarrow q(z)\prec p(z). $$ Applications and examples of the main results are also considered.

#### Article information

**Source**

Rocky Mountain J. Math., Volume 47, Number 2 (2017), 383-390.

**Dates**

First available in Project Euclid: 18 April 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.rmjm/1492502540

**Digital Object Identifier**

doi:10.1216/RMJ-2017-47-2-383

**Mathematical Reviews number (MathSciNet)**

MR3635364

**Zentralblatt MATH identifier**

1364.30017

**Subjects**

Primary: 30C45: Special classes of univalent and multivalent functions (starlike, convex, bounded rotation, etc.)

Secondary: 30C80: Maximum principle; Schwarz's lemma, Lindelöf principle, analogues and generalizations; subordination

**Keywords**

Convex univalent and starlike function strong differential superordination

#### Citation

Aghalary, R.; Arjomandinia, P.; Ebadian, A. Application of strong differential superordination to a general equation. Rocky Mountain J. Math. 47 (2017), no. 2, 383--390. doi:10.1216/RMJ-2017-47-2-383. https://projecteuclid.org/euclid.rmjm/1492502540