We study the weighted Hardy inequalities on the semiaxis of the form for functions vanishing at the endpoints together with derivatives up to the order . The case is completely characterized.

As is well known, invariant operators with a shift can be bounded from into only if . We show that the case might also hold for weighted spaces. We derive the sufficient conditions for the validity of strong (weak) type inequalities for the Hilbert transform when . The examples of couple of weights which guarantee the fulfillness of two-weighted strong (weak) type inequalities for ...

The main published inequality for Laguerre functions seems to be for Laguerre polynomials only; it is [2: 10.18(3)]: This paper presents several inequalities for Laguerre polynomials and Laguerre functions , most of which do not seem to be in the existing literature. The corresponding inequalities for confluent hypergeometric functions are noted. For our work on expansions in ...

We derive several inequalities for a functional connected with the well-known Jensen inequality on . Some applications for arithmetic and geometric means, and for the entropy mapping in information theory are also discussed.

The necessary and sufficient condition for equality to be attaind in the Kantrovich inequality is given and applied to an inequality of normal operators on Hilbert spaces.

Discrete version of Wirtinger's type inequality for higher differences, where and is considered. Under some conditions, the best constants , and are determined.

G. Bennett showed, by elementary proof, that if then (1.1) holds, and the constant is best possible; and if then (1.2) is valid. The reversed inequalities have remained open problems. As a first step into the converse direction, what seems to be very intricate without additional assumptions, we prove the inverse inequalities under slight restrictions on the monotonicity of the ...

The problem considered is to give necessary and sufficient conditions for perturbations of a second-order ordinary differential operator to be either relatively bounded or relatively compact. Such conditions are found for three classes of operators. The conditions are expressed in terms of integral averages of the coefficients of the perturbing operator.

Some inequalities will be presented, which give weighted norm estimates for derivatives of functions defined on the half-line. These inequalities are related to Hardy's inequality, and they also generalize Hardy's inequality to higher derivatives. The results presented here are also analogous to some recently-derived inequalities for the derivatives of functions defined on the ...

A characterization of a sharp form of Trudinger's inequality is established in terms of the Gagliardo-Nirenberg inequality in the limiting case for Sobolev's imbeddings.

For a function holomorphic and bounded, , with the expansion in the disk , we set Goluzin's extension of the Schwarz-Pick inequality is that We shall further improve Goluzin's inequality with a complete description on the equality condition. For a holomorphic map from a hyperbolic plane domain into another, one can prove a similar result in terms of the Poincaré metric.