Trapezoid inequality estimates the difference of integral mean a function on finite interval [a, b] and arithmetic its values at endpoints b. Quantum calculus is based diference principle or without concept limits. Euler-Jackson q-difference operator q-integral are discretization ordinary derivatives integrals they can be generalized to shifted versions arbitrary domain b]. In this paper we dis...

In this paper Lq → Lp boundedness of integral operator with operator-valued kernels is studied and main result is applied to convolution operators.

An elliptic Fourier integral operator of order 0, associated to a homogeneous canonical diffeomorphism, on a compact manifold is Fredholm on L2. The index may be expressed as the sum of a term, which we call the contact degree, associated to the canonical diffeomorphism and a term, computable by the Atiyah-Singer theorem, associated to the symbol. The contact degree is shown to be defined for a...

A new class of harmonic multivalent functions defined by an integral operator is introduced. Coefficient inequalities, extreme points, distortion bounds, inclusion results and closure under an integral operator for this class are obtained .

Let b ∈ BMO(Rn) and T be the Calderón–Zygmund singular integral operator. The commutator [b,T ] generated by b and T is defined as [b,T ]( f )(x) = b(x)T ( f )(x)−T (b f )(x). By using a classical result of Coifman et al [8], we know that the commutator [b,T ] is bounded on Lp(Rn) for 1 < p < ∞. Chanillo [1] proves a similar result when T is replaced by the fractional integral operator. However...