Ostrowski type inequalities for \(k\)-\(\beta\)-convex functions via Riemann-Liouville \(k\)-fractional integrals. (English) Zbl 1482.26023
A function \(f:I\subseteq \mathbb{R} \to \mathbb{R}\) is said to be \(k\)-\(\beta\)-convex on \(I\), if the inequality \[f(tx+(1-t)y) \le \frac{1}{k}t^{\frac{p}{k}}(1-t)^{\frac{q}{k}}f(x)+\frac{1}{k}t^{\frac{q}{k}}(1-t)^{\frac{p}{k}}f(y)\] holds for all \(x,y \in I\) and \(t \in [0,1],\) where \(p,q > -k,~k>0.\) In this paper, the author introduces a new concept of \(k\)-\(\beta\)-convex function to derive and prove some new Ostrowski-type inequalities for functions whose first derivatives in absolute value are \(k\)-\(\beta\)-convex via the concept of Riemann-Liouville \(k\)-fractional integrals.
Reviewer: James Adedayo Oguntuase (Abeokuta)
MSC:
| 26D10 | Inequalities involving derivatives and differential and integral operators |
| 26A33 | Fractional derivatives and integrals |
| 26A51 | Convexity of real functions in one variable, generalizations |
| 26D15 | Inequalities for sums, series and integrals |
Keywords:
Ostrowski inequality; Hölder inequality; power mean inequality; \(k\)-\(\beta\)-convex function; Riemann-Liouville \(k\)-fractional integralsReferences:
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