scholarly journals A variant of Jensen-type inequality and related results for harmonic convex functions

2020 ◽  
Vol 5 (6) ◽  
pp. 6404-6418 ◽  
Author(s):  
Imran Abbas Baloch ◽  
◽  
Aqeel Ahmad Mughal ◽  
Yu-Ming Chu ◽  
Absar Ul Haq ◽  
...  
Keyword(s):  
Convex Functions ◽  
Type Inequality ◽  
Harmonic Convex

scholarly journals Properties and Bounds of Jensen-Type Functionals via Harmonic Convex Functions

2021 ◽  
Vol 2021 ◽  
pp. 1-13
Author(s):  
Aqeel Ahmad Mughal ◽  
Hassan Almusawa ◽  
Absar Ul Haq ◽  
Imran Abbas Baloch
Keyword(s):  
Convex Functions ◽  
Functional Form ◽  
Type Inequality ◽  
Concave Function ◽  
Monotonicity Properties ◽  
Harmonic Convex

Dragomir introduced the Jensen-type inequality for harmonic convex functions (HCF) and Baloch et al. studied its different variants, such as Jensen-type inequality for harmonic h -convex functions. In this paper, we aim to establish the functional form of inequalities presented by Baloch et al. and prove the superadditivity and monotonicity properties of these functionals. Furthermore, we derive the bound for these functionals under certain conditions. Furthermore, we define more generalized functionals involving monotonic nondecreasing concave function as well as evince superadditivity and monotonicity properties of these generalized functionals.

scholarly journals Some Ostrowski type integral inequalities via generalized harmonic convex functions

2021 ◽  
Vol 5 (1) ◽  
pp. 200-208
Author(s):  
Muhammad Tariq ◽  
◽  
Saad Ihsan Butt ◽  
Keyword(s):  
Convex Functions ◽  
Type Inequality ◽  
Integral Inequalities ◽  
Ostrowski Type Inequality ◽  
Type Integral ◽  
Harmonic Convex

In this paper, we aim to introduce a new notion of convex functions namely the harmonic \(s\)-type convex functions. The refinements of Ostrowski type inequality are investigated which are the generalized and extended variants of the previously known results for harmonic convex functions.

scholarly journals A note on generalized convex functions

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Syed Zaheer Ullah ◽  
Muhammad Adil Khan ◽  
Yu-Ming Chu
Keyword(s):  
Convex Function ◽  
Convex Functions ◽  
Type Inequality ◽  
Generalized Convex Functions

Abstract In the article, we provide an example for a η-convex function defined on rectangle is not convex, prove that every η-convex function defined on rectangle is coordinate η-convex and its converse is not true in general, define the coordinate $(\eta _{1}, \eta _{2})$(η1,η2)-convex function and establish its Hermite–Hadamard type inequality.

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.

Refinements of the majorization-type inequalities via green and fink identities and related results

2018 ◽  
Vol 68 (4) ◽  
pp. 773-788 ◽  
Author(s):  
Sadia Khalid ◽  
Josip Pečarić ◽  
Ana Vukelić
Keyword(s):  
Green’S Function ◽  
Green's Function ◽  
Convex Functions ◽  
Type Inequality ◽  
Cauchy Type ◽  
Linear Functionals ◽  
Ostrowski’S Inequality ◽  
New Family ◽  
Exponentially Convex Functions ◽  
The Difference

Abstract In this work, the Green’s function of order two is used together with Fink’s approach in Ostrowski’s inequality to represent the difference between the sides of the Sherman’s inequality. Čebyšev, Grüss and Ostrowski-type inequalities are used to obtain several bounds of the presented Sherman-type inequality. Further, we construct a new family of exponentially convex functions and Cauchy-type means by looking to the linear functionals associated with the obtained inequalities.

scholarly journals On Sherman-Steffensen type inequalities

Filomat ◽  
2018 ◽  
Vol 32 (13) ◽  
pp. 4627-4638 ◽  
Author(s):  
Marek Niezgoda
Keyword(s):  
Convex Functions ◽  
Type Inequality ◽  
Special Cases ◽  
Steffensen Inequality

In this work, Sherman-Steffensen type inequalities for convex functions with not necessarily non-negative coefficients are established by using Steffensen?s conditions. The Brunk, Bellman and Olkin type inequalities are derived as special cases of the Sherman-Steffensen inequality. The superadditivity of the Jensen-Steffensen functional is investigated via Steffensen?s condition for the sequence of the total sums of all entries of the involved vectors of coeffecients. Some results of Baric et al. [2] and of Krnic et al. [11] on the monotonicity of the functional are recovered. Finally, a Sherman-Steffensen type inequality is shown for a row graded matrix.

General Harmonic Convex Functions and Integral Inequalities

2016 ◽  
pp. 443-472 ◽  
Author(s):  
Muhammad Aslam Noor ◽  
Khalida Inayat Noor ◽  
Muhammad Uzair Awan ◽  
Sabah Iftikhar
Keyword(s):  
Convex Functions ◽  
Integral Inequalities ◽  
Harmonic Convex

scholarly journals The left conformable fractional Hermite-Hadamard type inequalities for convex functions

Filomat ◽  
2019 ◽  
Vol 33 (8) ◽  
pp. 2417-2430
Author(s):  
Sercan Turhan
Keyword(s):  
Convex Functions ◽  
Fractional Integral ◽  
Type Inequality ◽  
Conformable Fractional Integral

In this paper, a new fractional Hermite-Hadamard type inequality for convex functions is obtained by using only the left conformable fractional integral. Also, to have new fractional trapezoid and midpoint type inequalities for the differentiable convex functions, two new equalities are proved.

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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