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
The aim of this paper is to present some comprehensive and extended versions of classical inequalities such as Radon's Inequality, Bergström's Inequality, the weighted power mean inequality, Schlömilch's Inequality and Nesbitt's Inequality on time scale calculus. In time scale calculus, results are unified and extended. The theory of time scale calculus is applied to unify discrete and continuous analysis and to combine them in one comprehensive form. This hybrid theory is also widely applied on dynamic inequalities. The study of dynamic inequalities has received a lot of attention in the literature and has become a major field in pure and applied mathematics.
In this research article, we investigate reverse Radon's inequality, reverse Bergström's inequality, the reverse weighted power mean inequality, reverse Schlömilch's inequality, reverse Bernoulli's inequality and reverse Lyapunov's inequality with Specht's ratio on time scales. We also present reverse Rogers--Holder's inequality with logarithmic mean and Specht's ratio on time scales. The time scale dynamic inequalities unify and extend some continuous inequalities and their corresponding discrete and quantum versions.
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