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THREE–PARAMETERWEIGHTEDHARDYTYPEINEQUALITIES B J M A BanachJ.Math.Anal.2(2008),no.2,85–93

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Banach J. Math. Anal. 2 (2008), no. 2, 85–93

B

anach

J

ournal of

M

athematical

A

nalysis ISSN: 1735-8787 (electronic)

http://www.math-analysis.org

THREE–PARAMETER WEIGHTED HARDY TYPE INEQUALITIES

R. OINAROV1∗ AND A. KALYBAY2 This paper is dedicated to Professor Josip E. Peˇcari´c

Submitted by S.-M. Jung

Abstract. For 0 < r < ∞ and 1 ≤ p ≤ q < ∞ we find necessary and sufficient conditions for the validity of the following inequality:

b

Z

a

u(x)

x

Z

a

|g(x)−g(t)|rw(t)dt

q r

dx

1 q

≤C

b

Z

a

v(x)|g0(x)|pdx

1 p

,

whereu(·),v(·), andw(·) are weight functions.

1 Eurasian National University, Munaytpasov st. 5, Astana 010008, Kaza- khstan.

E-mail address: o [email protected]

2 Kazakhstan Institute of Management, Economics and Strategic Research, Abay ave. 4, Almaty 050010, Kazakhstan.

E-mail address: aigerim [email protected]

Date: Received: 14 April 2008; Accepted: 18 June 2008.

∗ This research work was supported by the INTAS grant Ref. Nr 05-1000008-8157.

2000Mathematics Subject Classification. Primary 26D10, 26D15; Secondary 46E15, 46E30.

Key words and phrases. Inequalities, Hardy type inequalities, weight functions.

85

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2 Department of Mathematics, University of West Bohemia, Univerzitn´ı 22, 30614 Pilsen, Czech Republic.. E-mail