Abstract
We generalize Steffensen’s inequality to the class of n-convex functions using Taylor’s formula. Further, we use inequalities for the Cebysev functional to obtain bounds for identities related to generalizations of Steffensen’s inequality, and we give Ostrowski-type inequalities related to obtained generalizations. Finally, we apply our results to obtain new Stolarsky-type means.
Highlights
Remark . (i) Recall that ψ : R → C is entire if it can be extended to a necessarily unique analytic function ψ : C → C. (ii) Conclusions of Theorem . can be extended to exponentially convex functions defined on any open interval
We prove corollary of Theorem . which will be used to obtain new Stolarsky-type means
6 Stolarsky-type means we present several families of functions which fulfill the conditions of Theorem . , Corollary . and Corollary
Summary
In Steffensen proved the following inequality (see [ ]). Theorem. In this paper we generalize Steffensen’s inequality to n-convex functions using Taylor’s formula. We use inequalities for the Čebyšev functional to obtain bounds for identities related to generalizations of Steffensen’s inequality. Applying integration by parts and using the definition of the function G , identity ). In the following theorem we obtain generalizations of Steffensen’s inequality for nconvex functions. Let f : [a, b] → R be such that f is absolutely continuous, let g : [a, b] → R be an integrable function such that ≤ g ≤ and let λ =. ), and the remainder Hn (f ; a, b) satisfies the bound (iii) we have representation Taking n = in the previous theorem, we obtain the following corollary.
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
