Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Convex Holder bound and its applications

Hariprasad M

TL;DR

The paper introduces a convex Hölder bound that interpolates between pair of endpoint norms for l<s<m, with explicit exponents and a construction based on splitting |f|^s and applying Hölder. It provides a general bound and the m=∞ specialization, then demonstrates a Dinu-type inequality chain to motivate the approach. Three applications are developed: bounding odd zeta values, bounding binomial sums with non-integer exponents, and bounding gamma/beta-related integrals and functions via norm interpolation. Overall, the results yield sharp, implementable inequalities that connect endpoint norms to intermediate norms, with concrete consequences in analytic number theory and special-function estimates.

Abstract

Given l<s<m an upper bound on the s norm is given using l norm and m norm. The result is applied in bounding odd values of zeta function, binomial sums and gamma and beta functions.

Convex Holder bound and its applications

TL;DR

The paper introduces a convex Hölder bound that interpolates between pair of endpoint norms for l<s<m, with explicit exponents and a construction based on splitting |f|^s and applying Hölder. It provides a general bound and the m=∞ specialization, then demonstrates a Dinu-type inequality chain to motivate the approach. Three applications are developed: bounding odd zeta values, bounding binomial sums with non-integer exponents, and bounding gamma/beta-related integrals and functions via norm interpolation. Overall, the results yield sharp, implementable inequalities that connect endpoint norms to intermediate norms, with concrete consequences in analytic number theory and special-function estimates.

Abstract

Given l<s<m an upper bound on the s norm is given using l norm and m norm. The result is applied in bounding odd values of zeta function, binomial sums and gamma and beta functions.

Paper Structure

This paper contains 8 sections, 1 theorem, 21 equations, 1 table.

Table of Contents

  1. Introduction
  2. Results
  3. Dinu's inequality
  4. Convex Hölder bound
  5. Applications
  6. Bounding $\zeta(2k+1)$
  7. Bounding binomial sums
  8. Bounding integrals and special function

Key Result

Theorem 2.1

Given $l < s < m$ and given $l$ norm and $m$ norm of a function $f$ in a measure space, we have $(\norm{f}_s)^s \leq (\norm{f}_l)^{l\frac{(m-s)}{m-l}} (\norm{f}_m)^{m\frac{s-l}{m-l}}$.

Theorems & Definitions (2)

  • Theorem 2.1
  • proof