MathDB
Miklos Schweitzer 1971_9

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October 29, 2008
functionreal analysisreal analysis unsolved

Problem Statement

Given a positive, monotone function F(x) F(x) on (0,∞) (0, \infty) such that F(x)/x F(x)/x is monotone nondecreasing and F(x)/x1+d F(x)/x^{1+d} is monotone nonincreasing for some positive d d, let λn>0 \lambda_n >0 and an≥0,  n≥1 a_n \geq 0 , \;n \geq 1. Prove that if ∑n=1∞λnF(an∑k=1nλkλn)<∞, \sum_{n=1}^{\infty} \lambda_n F \left( a_n \sum _{k=1}^n \frac{\lambda_k}{\lambda_n} \right) < \infty, or ∑n=1∞λnF(∑k=1nakλkλn)<∞, \sum_{n=1}^{\infty} \lambda_n F \left( \sum _{k=1}^n a_k \frac{\lambda_k}{\lambda_n} \right) < \infty, then ∑n=1∞an \sum_{n=1}^ {\infty} a_n is convergent. L. Leindler
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