Popoviciu type inequalities via Green function and generalized Montgomery identity

Butt, Saad Ihsan; Khan, Khuram Ali; Pečarić, Josip

doi:10.7153/mia-18-118

Cited by 14 publications

(16 citation statements)

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“…In recent years many researchers have generalized the inequalities for m-convex functions; like S. I. Butt et al generalized the Popoviciu inequality for m-convex function using Taylor's formula, Lidstone polynomial, montgomery identity, Fink's identity, Abel-Gonstcharoff interpolation and Hermite interpolating polynomial (see [5][6][7][8][9]).…”

Section: Introduction and Preliminary Resultsmentioning

confidence: 99%

Estimation of Different Entropies via Taylor One Point and Taylor Two Points Interpolations Using Jensen Type Functionals

Niaz

Khan

Pec̆arić

et al. 2019

ijaa

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Section: Introduction and Preliminary Resultsmentioning

confidence: 99%

Estimation of Different Entropies via Taylor One Point and Taylor Two Points Interpolations Using Jensen Type Functionals

Niaz

Khan

Pec̆arić

et al. 2019

ijaa

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“…We use the idea given in [16]. In the sequel I and J are intervals in R. Proof The proof is similar to Theorem 4.6 in [4].…”

Section: Theorem 12 Letmentioning

confidence: 86%

“…It not only generalizes the results to higher-order convex functions but also extends the domain of interest from non-negative to real value (see [2][3][4]). The weight functions are also improved from positive to real weights.…”

Section: Introductionmentioning

confidence: 99%

Generalization of cyclic refinements of Jensen’s inequality by Fink’s identity

et al. 2018

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We generalize cyclic refinements of Jensen's inequality from a convex function to a higher-order convex function by means of Lagrange-Green's function and Fink's identity. We formulate the monotonicity of the linear functionals obtained from these identities utilizing the theory of inequalities for n-convex functions at a point. New Grüss-and Ostrowski-type bounds are found for identities associated with the obtained inequalities. Finally, we investigate the properties of linear functionals regarding exponential convexity and mean value theorems.MSC: Primary 26A51; 26D07; 26D15; 26D20; 26D99

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“…Corollary 1 gives (3) and therefore leads to (1), (2), and (4). Next we narrate some further important results of [4].…”

Section: Introductionmentioning

confidence: 83%

“…In [4], substitutions presented conclude that from (3) one may get all (1), (2), and (4). Recently, Fahad, Pečarić, and Praljak proved generalization [4] of (1) by extending the results given in [13].…”

Section: Introductionmentioning

confidence: 98%

Generalized Steffensen’s inequality by Montgomery identity

et al. 2019

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By using generalized Montgomery identity and Green functions we proved several identities which assist in developing connections with Steffensen's inequality. Under the assumptions of n-convexity and n-concavity many inequalities, which generalize Steffensen's inequality, inequalities from (Fahad et al. in their reverse, have been proved. Generalization of some inequalities (and their reverse) which are related to Hardy-type inequality (Fahad et al. in J. Math. Inequal. 9:481-487, 2015) have also been proved. New bounds of Ostrowski and Grüss type inequalities have been developed. Moreover, we formulate generalized Steffensen-type linear functionals and prove their monotonicity for the generalized class of (n + 1)-convex functions at a point. At the end, we present some applications of our study to the theory of exponentially convex functions. MSC: Primary 26D10; secondary 26D20Keywords: Steffensen's inequality; Green's function; Montgomery's identity; (n + 1)-convex function at a point c+θ c ψ(z) dz, where θ = d c f (z) dz.( 1 )A massive literature body dealing with several variants and improvements of (1) can be seen in [14,16] and the references therein. Pečarić [13] gave a nice generalization of (1) as follows.

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Popoviciu type inequalities via Green function and generalized Montgomery identity

Cited by 14 publications

References 0 publications

Estimation of Different Entropies via Taylor One Point and Taylor Two Points Interpolations Using Jensen Type Functionals

Estimation of Different Entropies via Taylor One Point and Taylor Two Points Interpolations Using Jensen Type Functionals

Generalization of cyclic refinements of Jensen’s inequality by Fink’s identity

Generalized Steffensen’s inequality by Montgomery identity

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