scholarly journals A class of strongly close-to-convex functions

2019 ◽  
Vol 38 (6) ◽  
pp. 9-24
Author(s):  
R. K. Raina ◽  
Poonam Sharma ◽  
Janusz Sokol
Keyword(s):  
Integral Operator ◽  
Convex Function ◽  
Unit Disk ◽  
Convex Functions ◽  
Coefficient Estimates ◽  
Special Cases ◽  
Radius Of Convexity

In this paper, we study a class of strongly close-to-convex functions $f(z)$ analytic in the unit disk $\mathbb{U}$ with $f(0)=0,f^{\prime }(0)=1$ satisfying for some convex function $g(z)$ the condition that\begin{equation*}\frac{zf^{\prime }(z)}{g(z)}\prec \left( \frac{1+Az}{1+Bz}\right) ^{m}\end{equation*}%\begin{equation*}\left( -1\leq A\leq 1,-1\leq B\leq 1\ \left( A\neq B\right) ,0<m\leq 1;z\in\mathbb{U}\right) .\end{equation*}%We obtain for functions belonging to this class, the coefficient estimates, bounds, certain results based on an integral operator and radius of convexity. We also deduce a number of useful special cases and consequences of the various results which are presented in this paper.

scholarly journals A Generalized Fejér-Hadamard Inequality for Harmonically Convex Functions via Generalized Fractional Integral Operator and Related Results

2018 ◽  
Author(s):  
Shin Min Kang ◽  
Ghulam Abbas ◽  
Ghulam Farid ◽  
Waqas Nazeer
Keyword(s):  
Integral Operator ◽  
Convex Function ◽  
Convex Functions ◽  
Fractional Integral ◽  
Fractional Integral Operator ◽  
Integral Inequalities ◽  
Hadamard Inequality ◽  
Special Cases ◽  
Generalized Fractional Integral ◽  
Harmonically Convex Functions

In the present research, we will develop some integral inequalities of Hermite Hadamard type for differentiable &eta;-convex function. Moreover, our results include several new and known results as special cases.

On a subclass of quasi-convex functions with fixed second coefficient

2017 ◽  
Vol 10 (04) ◽  
pp. 1750078 ◽  
Author(s):  
G. Thirupathi
Keyword(s):  
Convex Function ◽  
Unit Disk ◽  
Convex Functions ◽  
Analytic Functions ◽  
Radius Of Convexity ◽  
Distortion Theorems ◽  
Growth And Distortion Theorems

In this paper, we introduce the concept of quasi-convex functions by considering a subclass [Formula: see text] of normalized analytic functions in the unit disk [Formula: see text] for some convex function [Formula: see text] with fixed second coefficient of order [Formula: see text], such that [Formula: see text] We obtain results on growth and distortion theorems and radius of convexity.

scholarly journals Subordination by convex functions

2006 ◽  
Vol 2006 ◽  
pp. 1-6 ◽  
Author(s):  
Rosihan M. Ali ◽  
V. Ravichandran ◽  
N. Seenivasagan
Keyword(s):  
Analytic Function ◽  
Convex Function ◽  
Unit Disk ◽  
Convex Functions ◽  
Analytic Functions ◽  
Open Unit ◽  
Open Unit Disk ◽  
Subordination Result ◽  
Special Cases

For a fixed analytic functiong(z)=z+∑n=2∞gnzndefined on the open unit disk andγ<1, letTg(γ)denote the class of all analytic functionsf(z)=z+∑n=2∞anznsatisfying∑n=2∞|angn|≤1−γ. For functions inTg(γ), a subordination result is derived involving the convolution with a normalized convex function. Our result includes as special cases several earlier works.

scholarly journals Inequalities for a Unified Integral Operator for Strongly α , m -Convex Function and Related Results in Fractional Calculus

2021 ◽  
Vol 2021 ◽  
pp. 1-8
Author(s):  
Chahn Yong Jung ◽  
Ghulam Farid ◽  
Kahkashan Mahreen ◽  
Soo Hak Shim
Keyword(s):  
Integral Operator ◽  
Fractional Calculus ◽  
Convex Function ◽  
Convex Functions ◽  
Fractional Integral ◽  
Integral Operators ◽  
Integral Inequalities ◽  
Special Cases ◽  
Definition Of

In this paper, we study integral inequalities which will provide refinements of bounds of unified integral operators established for convex and α , m -convex functions. A new definition of function, namely, strongly α , m -convex function is applied in different forms and an extended Mittag-Leffler function is utilized to get the required results. Moreover, the obtained results in special cases give refinements of fractional integral inequalities published in this decade.

scholarly journals New Modifications of Integral Inequalities via ℘-Convexity Pertaining to Fractional Calculus and Their Applications

Mathematics ◽  
2021 ◽  
Vol 9 (15) ◽  
pp. 1753
Author(s):  
Saima Rashid ◽  
Aasma Khalid ◽  
Omar Bazighifan ◽  
Georgia Irina Oros
Keyword(s):  
Integral Operator ◽  
Convex Functions ◽  
Hypergeometric Functions ◽  
Fractional Integral ◽  
Type Inequality ◽  
Fractional Integral Operator ◽  
Integral Inequalities ◽  
Convex Mappings ◽  
Special Cases ◽  
Harmonically Convex Functions

Integral inequalities for ℘-convex functions are established by using a generalised fractional integral operator based on Raina’s function. Hermite–Hadamard type inequality is presented for ℘-convex functions via generalised fractional integral operator. A novel parameterized auxiliary identity involving generalised fractional integral is proposed for differentiable mappings. By using auxiliary identity, we derive several Ostrowski type inequalities for functions whose absolute values are ℘-convex mappings. It is presented that the obtained outcomes exhibit classical convex and harmonically convex functions which have been verified using Riemann–Liouville fractional integral. Several generalisations and special cases are carried out to verify the robustness and efficiency of the suggested scheme in matrices and Fox–Wright generalised hypergeometric functions.

scholarly journals New Inequalities for Strongly r-Convex Functions

2019 ◽  
Vol 2019 ◽  
pp. 1-10 ◽  
Author(s):  
Huriye Kadakal
Keyword(s):  
Convex Function ◽  
Convex Functions ◽  
Integral Identity ◽  
Integral Inequalities ◽  
Special Cases ◽  
Class Of Functions ◽  
New Inequalities

In this study, firstly we introduce a new concept called “strongly r-convex function.” After that we establish Hermite-Hadamard-like inequalities for this class of functions. Moreover, by using an integral identity together with some well known integral inequalities, we establish several new inequalities for n-times differentiable strongly r-convex functions. In special cases, the results obtained coincide with the well-known results in the literature.

scholarly journals Some Connections between Class𝒰- andα-Convex Functions

2014 ◽  
Vol 2014 ◽  
pp. 1-4
Author(s):  
Edmond Aliaga ◽  
Nikola Tuneski
Keyword(s):  
Convex Function ◽  
Unit Disk ◽  
Convex Functions ◽  
Analytic Functions ◽  
Sufficient Conditions ◽  
Open Unit ◽  
Open Unit Disk

The class𝒰(λ,μ)of normalized analytic functions that satisfy|(z/f(z))1+μ·f′(z)−1|<λfor allzin the open unit disk is studied and sufficient conditions for anα-convex function to be in𝒰(λ,μ)are given.

scholarly journals Geometric Properties of a New Integral Operator

2015 ◽  
Vol 2015 ◽  
pp. 1-6
Author(s):  
Roberta Bucur ◽  
Loriana Andrei ◽  
Daniel Breaz
Keyword(s):  
Integral Operator ◽  
Unit Disk ◽  
Analytic Functions ◽  
Sufficient Conditions ◽  
Open Unit ◽  
Open Unit Disk ◽  
Geometric Properties ◽  
Special Cases

We obtain sufficient conditions for the univalence, starlikeness, and convexity of a new integral operator defined on the space of normalized analytic functions in the open unit disk. Some subordination results for the new integral operator are also given. Several corollaries follow as special cases.

scholarly journals Some Hermite-Hadamard type inequalities for (P;m)-function and quasi m-convex functions

2020 ◽  
Vol 10 (1) ◽  
pp. 78-84
Author(s):  
Mahir Kadakal
Keyword(s):  
Convex Function ◽  
Convex Functions ◽  
New Class ◽  
Special Cases ◽  
First Time ◽  
Class Of Functions

In this paper, we introduce a new class of functions called as (P;m)-function and quasi-m-convex function. Some inequalities of Hadamard's type for these functions are given. Some special cases are discussed. Results represent signicant renement and improvement of the previous results. We should especially mention that the denition of (P;m)-function and quasi-m-convexity are given for the first time in the literature and moreover, the results obtained in special cases coincide with thewell-known results in the literature.

scholarly journals Some inequalities for strongly $(p,h)$-harmonic convex functions

2019 ◽  
Vol 11 (1) ◽  
pp. 119-135
Author(s):  
M.A. Noor ◽  
K.I. Noor ◽  
S. Iftikhar
Keyword(s):  
Convex Function ◽  
Convex Functions ◽  
New Class ◽  
Special Cases ◽  
Harmonic Convex

In this paper, we show that harmonic convex functions $f$ is strongly $(p, h)$-harmonic convex functions if and only if it can be decomposed as $g(x) = f(x) - c (\frac{1}{x^p})^2,$ where $g(x)$ is $(p, h)$-harmonic convex function. We obtain some new estimates  class of strongly $(p, h)$-harmonic convex functions involving hypergeometric and beta functions. As applications of our results, several important special cases are discussed. We also introduce a new class of harmonic convex functions, which is called strongly $(p, h)$-harmonic $\log$-convex functions. Some new Hermite-Hadamard type inequalities for strongly $(p, h)$-harmonic $log$-convex functions are obtained. These results  can be viewed as important refinement and significant improvements of the new and previous known results. The ideas and techniques of this paper may stimulate further research.

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