Dynamic Inequalities for Convex Functions Harmonized on Time Scales

Abstract

We present here some general fractional Schlömilch's type and Rogers-Hölder's type dynamic inequalities for convex functions harmonized on time scales. First we present general fractional Schlömilch's type dynamic inequalities and generalize it for convex functions of several variables by using Bernoulli's inequality, generalized Jensen's inequality and Fubini's theorem on diamond-α calculus. To conclude our main results, we present general fractional Rogers-Hölder's type dynamic inequalities for convex functions by using general fractional Schlömilch's type dynamic inequality on diamond-α calculus for

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“…Over the previous decade, a reasonable number of dynamic inequalities on time scales has been proven by many analysts who were propelled by certain applications (see [1-4, 9-14, 18, 29]). A few researchers created different outcomes concerning fractional calculus on time scales to deliver related dynamic inequalities (see [5][6][7]24]).…”

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“…Inequality (37) is called Schlömilch's inequality by using the time scale Riemann-Liouville type fractional integral. Its other versions are also given in [16,18,21].…”

Dynamic Fractional Inequalities Amplified on Time Scale Calculus Revealing Coalition of Discreteness and Continuity

“…The inequality given in upcoming corollary is called Schlömilch's inequality. Its other versions are also given in [16,17].…”