On Some Differential Inequalities with Applications

International Journal of Analysis, 2015

LetAbe the class of analytic functionsfdefined in the open unit diskEand normalized byf(0)=f'(0)-1=0. Forf(z)/z≠0inE, letMα,λ:={f∈A:|-αz2(z/f(z))′′ +f′(z)(z/f(z))2-1|≤λ, z∈E}, whereλ>0andα∈R∖[-1/2,0]. In the present paper, we find conditions under which functions in the classM(α,λ)are starlike of orderγ,0≤γ<1.

Let Bp(α, β, λ; j) be the class consisting of functions f (z) = z p + ∑ ∞ k=p+1 a k z k , p ∈ N which satisfy Re { α f (j) (z) z p−j + β f (j+1) (z) z p−j−1 + (β − α 2) f (j+2) (z) z p−j−2 } > λ, (z ∈ U = {z : |z| < 1}), for some λ (λ < p!{α+(p−j)β +(p−j)(p−j −1)(β −α)/2}/(p−j)!) and j = 0, 1, ..., p , where p+1−j +2α/(β −α) > 0 or α = β = 1. The extreme points of Bp(α, β, λ; j) are determined and various sharp inequalities related to Bp(α, β, λ; j) are obtained. These include univalence criteria, coefficient bounds, growth and distortion estimates and bounds for certain linear operators. Furthermore, inclusion properties are investigated and estimates on λ are found so that functions of Bp(α, β, λ; j) are p-valent starlike in U. For instance, Re{zf ′′ (z)} > (5 − 12 ln 2)/(44 − 48 ln 2) ≈ −0.309 is sufficient condition for any normalized analytic function f to be starlike in U. The results improve and include a number of known results as their special cases.

Journal of Inequalities in Pure and Applied Mathematics

Some interesting inequalities proved by Dragomir and van der Hoek are generalized with some remarks on the results.

Advances in Difference Equations, 2021

In this paper, by using a technique of the first-order differential subordination, we find several sufficient conditions for an analytic function p such that $p(0)=1$ p ( 0 ) = 1 to satisfy $\operatorname{Re}\{ {\mathrm{e}}^{{\mathrm{i}}\beta } p(z) \} &gt; \gamma $ Re { e i β p ( z ) } &gt; γ or $| \arg \{p(z)-\gamma \} |&lt;\delta $ | arg { p ( z ) − γ } | &lt; δ for all $z\in \mathbb{D}$ z ∈ D , where $\beta \in (-\pi /2,\pi /2)$ β ∈ ( − π / 2 , π / 2 ) , $\gamma \in [0,\cos \beta )$ γ ∈ [ 0 , cos β ) , $\delta \in (0,1]$ δ ∈ ( 0 , 1 ] and $\mathbb{D}:=\{z\in \mathbb{C}:|z|&lt;1 \}$ D : = { z ∈ C : | z | &lt; 1 } . The results obtained here will be applied to find some conditions for spirallike functions and strongly starlike functions in $\mathbb{D}$ D .

In the present paper, using Jack's lemma, the authors investigate the differential inequality

The object of the present paper is to give an application of the fractional derivative operator for p-valent functions in the open unit disk to the differential inequalities.

It is proved that if a linear operator l : C((a;b);R)!L((a;b);R) is nonpositive and for the Cauchy problem u00(t) = l(u)(t) + q(t), u(a) = c the theorem on dierential inequalities is valid, then l is a Volterra operator.

Journal of Inequalities and Applications, 2012

The aim of this paper is to study the properties of a subclass of analytic functions related to p-valent Bazilevic functions by using the concept of differential subordination. We investigate some results concerned with coefficient bounds, inclusion results, radius problem, covering theorem, angular estimation of a certain integral operator, and some other interesting properties. MSC: 30C45; 30C50

Acta Universitatis Sapientiae, Mathematica, 2018

In this note, an extensive result consisting of several relations between certain inequalities and normalized analytic functions is first stated and some consequences of the result together with some examples are next presented. For the proof of the presented result, some of the assertions indicated in [5], [8] and [11] along with the results in [3] and [4] are also considered.

emis.ams.org

In the present paper, the authors investigate a differential inequality defined by multiplier transformation in the open unit disk E = {z : |z| < 1}. As consequences, sufficient conditions for starlikeness and convexity of analytic functions are obtained.