Certain Hermite–Hadamard Inequalities for Logarithmically Convex Functions with Applications

Jain, Shilpi; Mehrez, Khaled; Băleanu, Dumitru; Agarwal, Praveen

doi:10.3390/math7020163

Cited by 39 publications

(29 citation statements)

References 7 publications

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“…4 Institute of Space Sciences, Magurele, 077125 Bucharest, Romania. 5 Department of Mathematics, Huzhou University, 313000 Huzhou, P.R. China.…”

Section: Lemma 20mentioning

confidence: 99%

“…Sitho et al [4] presented some new non-instantaneous impulsive inequalities us-ing the conformable fractional calculus. Jain et al [5] discussed various estimates by the Hermite-Hadamard inequality for functions whose absolute values of the second derivatives to positive real powers are log-convex. Tomar et al [6] established some generalized Hermite-Hadamard inequalities for generalized convex functions.…”

Section: Introductionmentioning

confidence: 99%

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A divided differences based medium to analyze smoothness of the binary bivariate refinement schemes

et al. 2021

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In this article, we present the continuity analysis of the 3D models produced by the tensor product scheme of $(m+1)$ ( m + 1 ) -point binary refinement scheme. We use differences and divided differences of the bivariate refinement scheme to analyze its smoothness. The $C^{0}$ C 0 , $C^{1}$ C 1 and $C^{2}$ C 2 continuity of the general bivariate scheme is analyzed in our approach. This gives us some simple conditions in the form of arithmetic expressions and inequalities. These conditions require the mask and the complexity of the given refinement scheme to analyze its smoothness. Moreover, we perform several experiments by using these conditions on established schemes to verify the correctness of our approach. These experiments show that our results are easy to implement and are applicable for both interpolatory and approximating types of the schemes.

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“…4 Institute of Space Sciences, Magurele, 077125 Bucharest, Romania. 5 Department of Mathematics, Huzhou University, 313000 Huzhou, P.R. China.…”

Section: Lemma 20mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A divided differences based medium to analyze smoothness of the binary bivariate refinement schemes

et al. 2021

Self Cite

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“…The study of fractional calculus, which involves fractional derivatives and integrals, has allured the interest of many in the field of engineering and natural sciences due to its monumental applications such as found in biotechnology [1], chaos theory [2], electrodynamics [3], random walk [4], signal and image processing [5,6], nanotechnology [7], viscoelasticity [8], and other various fields [9][10][11][12][13][14][15][16][17][18]. We also refer the reader to [19][20][21][22][23][24][25][26] for some recent applications of fractional calculus. Many researchers have also described the essential properties of this fractional calculus, see [27][28][29][30] for more details.…”

Section: Introductionmentioning

confidence: 99%

A reliable technique to study nonlinear time-fractional coupled Korteweg–de Vries equations

Akinyemi

Iyiola

2020

Adv Differ Equ

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This paper employs an efficient technique, namely q-homotopy analysis transform method, to study a nonlinear coupled system of equations with Caputo fractional-time derivative. The nonlinear fractional coupled systems studied in this present investigation are the generalized Hirota-Satsuma coupled with KdV, the coupled KdV, and the modified coupled KdV equations which are used as a model in nonlinear physical phenomena arising in biology, chemistry, physics, and engineering. The series solution obtained using this method is proved to be reliable and accurate with minimal computations. Several numerical comparisons are made with well-known analytical methods and the exact solutions when α = 1. It is evident from the results obtained that the proposed method outperformed other methods in handling the coupled systems considered in this paper. The effect of the fractional order on the problem considered is investigated, and the error estimate when compared with exact solution is presented.Here, we present some useful definitions, properties, and notations that will be used in this work. Definition 2.1 The Riemann-Liouville (R-L) fractional integral of order α (α ≥ 0) of a function Q(x, t) ∈ C m , m ≥ -1, is given as [29, 63-65] J α Q(x, t) = 1 Γ (α) t 0

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“…Moreover, convex functions have some nice properties, such as differentiability, monotonicity, and continuity, which are useful in applications [1][2][3][4][5]. Interest in mathematical inequalities for convex and generalized convex functions has been growing exponentially, and research in this respect has had a significant impact on modern analysis [6][7][8][9][10][11][12][13][14][15][16][17][18][19][20]. Several mathematical inequalities have been established for s-convex functions in particular [21][22][23][24][25][26][27][28], one of the most important being the Jensen inequality.…”

Section: Introductionmentioning

confidence: 99%

New Estimates for the Jensen Gap Using s-Convexity With Applications

Khan

Chu

2020

Front. Phys.

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In this article, we use s-convex and Green functions to obtain a bound for the Jensen gap in discrete form and a bound for the Jensen gap in integral form. We present two numerical examples to verify the main results and to examine the tightness of the bounds. Then, as an application of the discrete result, we derive a converse of the Hölder inequality. Based on the integral result, we obtain a bound for the Hermite-Hadamard gap and present a converse of the Hölder inequality in its integral form. Also, we obtain bounds for the Csiszár and Rényi divergences as applications of the discrete result. Finally, we utilize the bound obtained for the Csiszár divergence to deduce new estimates for some other divergences in information theory.

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Certain Hermite–Hadamard Inequalities for Logarithmically Convex Functions with Applications

Cited by 39 publications

References 7 publications

A divided differences based medium to analyze smoothness of the binary bivariate refinement schemes

A divided differences based medium to analyze smoothness of the binary bivariate refinement schemes

A reliable technique to study nonlinear time-fractional coupled Korteweg–de Vries equations

New Estimates for the Jensen Gap Using s-Convexity With Applications

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