Generalized Steffensen's inequality

Fahad, Asfand; Pečarić, Josip; Praljak, Marjan

doi:10.7153/jmi-09-41

Cited by 5 publications

(19 citation statements)

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“…Recently, Fahad, Pečarić, and Praljak proved generalization [4] of (1) by extending the results given in [13]. Now, we include some important conclusions given in [4].…”

Section: Introductionmentioning

confidence: 55%

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

Section: Introductionmentioning

confidence: 96%

“…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]. Now, we include some important conclusions given in [4].…”

Section: Introductionmentioning

confidence: 97%

“…Integral inequalities such as Hardy's inequality, Steffensen's inequality, and Ostrowski's inequality have been topics of interest of many mathematicians since their pronouncement. Several generalizations of these inequalities have been proved for convex functions, beta m-convex functions, n-convex functions, and other classes of functions, for example, see [4,7,12,16], and [17]. Steffensen's inequality was proved in [18]: if ψ, f : [c, d] → R, with ψ being a decreasing function and function f having range in [0, 1], then…”

Section: Introductionmentioning

confidence: 99%

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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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“…Recently, Fahad, Pečarić, and Praljak proved generalization [4] of (1) by extending the results given in [13]. Now, we include some important conclusions given in [4].…”

Section: Introductionmentioning

confidence: 55%

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

Section: Introductionmentioning

confidence: 96%

Section: Introductionmentioning

confidence: 97%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Generalized Steffensen’s inequality by Montgomery identity

et al. 2019

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“…Hermite interpolation of composition function and some related inequalities are given. The obtained inequalities are closely related to a generalization of Steffensen's inequality given by the current authors in [2].…”

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confidence: 99%