ABSTRACT
In their volumes Problems and Theorems in Analysis, Pólya and Szegö [32] presented the following related results.
Theorem A Let f : [0, 1] → R be a nonnegative and increasing function. If a and b are nonnegative real numbers, then ( ∫ 0 1 x a + b f ( x ) d x ) 2 ≥ [ 1 − ( a − b a + b + 1 ) 2 ] ∫ 0 1 x 2 a f ( x ) d x ∫ 0 1 x 2 b f ( x ) d x . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429123610/50e9be86-7107-4365-8f7e-f2f1821b70e3/content/eq2846.tif"/>
Theorem B Let f : [0, ∞) → R be a nonnegative and decreasing function. If a and b are nonnegative real numbers, then [ ∫ 0 ∞ x a + b f ( x ) d x ] 2 ≤ [ 1 − ( a − b a + b + 1 ) 2 ] ∫ 0 ∞ x 2 a f ( x ) d x ∫ 0 ∞ x 2 b f ( x ) d x . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429123610/50e9be86-7107-4365-8f7e-f2f1821b70e3/content/eq2847.tif"/>
466These inequalities have since been improved, generalized and applied in a number of articles. There are several distinct streams of generalization. In one the class of weighted function f is broadened, in another the interval of integration extended. There are results in which the power functions x a+b , x 2a and x 2b are replaced by more general functions. One line of generalization leads to different kinds of means. Discrete Pólya-type inequalities have also been obtained. Beyond all of this, techniques for the real scalar case have been adapted to operator theory. As a consequence, a number of results for positive linear operators have been derived.
In this chapter we give an account of this fertile and rapidly growing field, including general ideas, characteristic proofs and applications to probability theory.
