scholarly journals Determination of Bounds for the Jensen Gap and Its Applications

Mathematics ◽  
2021 ◽  
Vol 9 (23) ◽  
pp. 3132
Author(s):  
Hidayat Ullah ◽  
Muhammad Adil Khan ◽  
Tareq Saeed
Keyword(s):  
Information Theory ◽  
Current Literature ◽  
Convex Functions ◽  
Concave Function ◽  
Hölder Inequality ◽  
Jensen Inequality ◽  
Power Mean ◽  
Arithmetic Means ◽  
Jensen Difference

The Jensen inequality has been reported as one of the most consequential inequalities that has a lot of applications in diverse fields of science. For this reason, the Jensen inequality has become one of the most discussed developmental inequalities in the current literature on mathematical inequalities. The main intention of this article is to find some novel bounds for the Jensen difference while using some classes of twice differentiable convex functions. We obtain the proposed bounds by utilizing the power mean and Höilder inequalities, the notion of convexity and the prominent Jensen inequality for concave function. We deduce several inequalities for power and quasi-arithmetic means as a consequence of main results. Furthermore, we also establish different improvements for Hölder inequality with the help of obtained results. Moreover, we present some applications of the main results in information theory.

scholarly journals Refinements of Jensen's inequality and applications

2022 ◽  
Vol 7 (4) ◽  
pp. 5328-5346
Author(s):  
Tareq Saeed ◽  
◽  
Muhammad Adil Khan ◽  
Hidayat Ullah ◽  
Keyword(s):  
Information Theory ◽  
Shannon Entropy ◽  
Research Work ◽  
Jensen’S Inequality ◽  
Hölder Inequality ◽  
Jensen's Inequality ◽  
Arithmetic Means ◽  
Hadamard Inequality ◽  
Bhattacharyya Coefficient

<abstract><p>The principal aim of this research work is to establish refinements of the integral Jensen's inequality. For the intended refinements, we mainly use the notion of convexity and the concept of majorization. We derive some inequalities for power and quasi–arithmetic means while utilizing the main results. Moreover, we acquire several refinements of Hölder inequality and also an improvement of Hermite–Hadamard inequality as consequences of obtained results. Furthermore, we secure several applications of the acquired results in information theory, which consist bounds for Shannon entropy, different divergences, Bhattacharyya coefficient, triangular discrimination and various distances.</p></abstract>

scholarly journals Some Inequalities for a New Class of Convex Functions with Applications via Local Fractional Integral

2021 ◽  
Vol 2021 ◽  
pp. 1-17
Author(s):  
Hu Ge-JiLe ◽  
Saima Rashid ◽  
Fozia Bashir Farooq ◽  
Sobia Sultana
Keyword(s):  
Applied Sciences ◽  
Convex Functions ◽  
Fractional Integral ◽  
Hölder Inequality ◽  
Power Mean ◽  
New Class ◽  
Local Fractional Calculus ◽  
Local Fractional Integral ◽  
User Friendly ◽  
Harmonically Convex Functions

The understanding of inequalities in convexity is crucial for studying local fractional calculus efficiency in many applied sciences. In the present work, we propose a new class of harmonically convex functions, namely, generalized harmonically ψ - s -convex functions based on fractal set technique for establishing inequalities of Hermite-Hadamard type and certain related variants with respect to the Raina’s function. With the aid of an auxiliary identity correlated with Raina’s function, by generalized Hölder inequality and generalized power mean, generalized midpoint type, Ostrowski type, and trapezoid type inequalities via local fractional integral for generalized harmonically ψ - s -convex functions are apprehended. The proposed technique provides the results by giving some special values for the parameters or imposing restrictive assumptions and is completely feasible for recapturing the existing results in the relative literature. To determine the computational efficiency of offered scheme, some numerical applications are discussed. The results of the scheme show that the approach is straightforward to apply and computationally very user-friendly and accurate.

scholarly journals Extensions of recent combinatorial refinements of discrete and integral Jensen inequalities

2021 ◽  
Author(s):  
László Horváth
Keyword(s):  
Information Theory ◽  
Geometric Mean ◽  
Arithmetic Mean ◽  
Jensen Inequality ◽  
Arithmetic Means ◽  
Norm Inequalities ◽  
Integrable Functions ◽  
Essential Extensions ◽  
Vector Valued

AbstractThe main purpose of this work is to present essential extensions of results in [7] and [8], and apply them to some special situations. Of particular interest is the refinement of the integral Jensen inequality for vector valued integrable functions. The applications related to four topics, namely f-divergences in information theory (an interesting refinement of the weighted geometric mean–arithmetic mean inequality is obtained as a consequence), norm inequalities, quasi-arithmetic means, Hölder’s and Minkowski’s inequalities.

scholarly journals SOME REVERSES OF THE JENSEN INEQUALITY WITH APPLICATIONS

2013 ◽  
Vol 87 (2) ◽  
pp. 177-194 ◽  
Author(s):  
S. S. DRAGOMIR
Keyword(s):  
Information Theory ◽  
Convex Functions ◽  
Hölder’S Inequality ◽  
General Setting ◽  
Jensen’S Inequality ◽  
Jensen Inequality ◽  
Lebesgue Integral ◽  
Jensen's Inequality ◽  
Hölder's Inequality ◽  
Divergence Measures

AbstractTwo new reverses of the celebrated Jensen’s inequality for convex functions in the general setting of the Lebesgue integral, with applications to means, Hölder’s inequality and$f$-divergence measures in information theory, are given.

scholarly journals A New Information Inequality and Its Application in Establishing Relation Among Various f-Divergence Measures

2012 ◽  
Vol 8 (1) ◽  
pp. 17-32 ◽  
Author(s):  
K. Jain ◽  
Ram Saraswat
Keyword(s):  
Information Theory ◽  
Jensen Inequality ◽  
Chi Square ◽  
Information Inequality ◽  
Divergence Measures ◽  
New Information

A New Information Inequality and Its Application in Establishing Relation Among Various f-Divergence MeasuresAn Information inequality by using convexity arguments and Jensen inequality is established in terms of Csiszar f-divergence measures. This inequality is applied in comparing particular divergences which play a fundamental role in Information theory, such as Kullback-Leibler distance, Hellinger discrimination, Chi-square distance, J-divergences and others.

scholarly journals The Weighted Arithmetic Mean–Geometric Mean Inequality is Equivalent to the Hölder Inequality

Symmetry ◽  
2018 ◽  
Vol 10 (9) ◽  
pp. 380 ◽  
Author(s):  
Yongtao Li ◽  
Xian-Ming Gu ◽  
Jianxing Zhao
Keyword(s):  
Geometric Mean ◽  
Arithmetic Mean ◽  
Hölder Inequality ◽  
Power Mean ◽  
Weighted Arithmetic ◽  
Mathematical Equivalence ◽  
Power Mean Inequality ◽  
Mathematical Relations

In the current note, we investigate the mathematical relations among the weighted arithmetic mean–geometric mean (AM–GM) inequality, the Hölder inequality and the weighted power-mean inequality. Meanwhile, the proofs of mathematical equivalence among the weighted AM–GM inequality, the weighted power-mean inequality and the Hölder inequality are fully achieved. The new results are more generalized than those of previous studies.

Fracture Resistance of AAAS Composites with Ceramic Precursor

2021 ◽  
Vol 322 ◽  
pp. 54-59
Author(s):  
Iva Rozsypalová ◽  
Petr Daněk ◽  
Pavla Rovnaníková ◽  
Zbyněk Keršner
Keyword(s):  
Mouth Opening ◽  
Crack Model ◽  
Arithmetic Means ◽  
Mechanical Fracture ◽  
Three Point Bending ◽  
Static Modulus ◽  
Fracture Parameters ◽  
Ceramic Precursors ◽  
Static Modulus Of Elasticity

The paper deals with selected alkali-activated aluminosilicate (AAAS) composites based on ceramic precursors in terms of their characterization by mechanical fracture parameters. Composites made of brick dust as a precursor and an alkaline activator with a silicate modulus of Ms = 0.8, 1.0, 1.2, 1.4 and 1.6 were investigated. The filler employed with one set of composites was quartz sand, while for the other set it was crushed brick. The test specimens had nominal dimensions of 40 × 40 × 160 mm and were provided with notches at midspan of up to 1/3 of the height of the specimens after 28 days. 6 samples from each composite were tested. The specimens were subjected to three-point bending tests in which force vs. displacement (deflection at midspan) diagrams (F–d diagrams) and force vs. crack mouth opening (F–CMOD) diagrams were recorded. After the correction of these diagrams, static modulus of elasticity, effective fracture toughness, effective toughness and specific fracture energy values were determined using the Effective Crack Model and the Work-of-Fracture method. After the fracture experiments, informative compressive strength values were determined from one of the parts. All of the evaluations included the determination of arithmetic means and standard deviations. The silicate modulus values and type of filler of the AAAS composites significantly influenced their mechanical fracture parameters.

The Early History of the Gyroscope and Gyro Compass

1970 ◽  
Vol 24 (1) ◽  
pp. 86-89
Author(s):  
G. B. Lauf
Keyword(s):  
Current Literature ◽  
Early History ◽  
Historical Account ◽  
Half Century ◽  
History Of ◽  
The Subject ◽  
Gyro Compass

Most of the current literature in the field of gyroscopic theory and in the use of gyroscopic instruments for the determination of azimuth begins the historical account of the subject with the work of Leon Foucault during the period 1850-1852. But little is known of the work in this field by others during the preceding half century. In this paper, the development of the gyroscope and gyro compass is traced back to a date earlier than 1813.

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