A family of integral equations with restricted solutions

Abstract

A family of Volterra integral equations of a special kind is considered. In each equation a given function occurs both in the integrand and in the expression on the right equal to the convolution integral. It is proved that all solutions to the equations of this family satisfy certain sharp mean modulus inequality. This result can be used as a source of various inequalities for special functions. Some examples and applications that are based on the explicit hypergeometric and general series solutions are discussed. They involve the Jacobi and Laguerre polynomials as well as the confluent hypergeometric functions.

Keywords:

• Volterra equations of the first kind
• convolutions
• weighted integral inequalities
• hypergeometric functions
• Jacobi polynomials
• Laguerre polynomials

2000 Mathematics Subject Classification :

• 26D15
• 45D05
• 33B15
• 33C05
• 33C15
• 33C45