An integral inequality for cosine polynomials

摘要

LetTn(x)=12a0+∑k=1nakcos(kx).We prove that if ak (k=0,1,…,n) is an increasing sequence of real numbers, then for any function f which is increasing and convex on the real line we have1π∫0πf|Tn(x)|Lndx⩾f1n+1∑k=0nak,whereLn=1π∫0π12+∑k=1ncos(kx)dxdenotes the Lebesgue constant. This extends and refines a result due to Fejes (1939).