In this note, first the better refinements of Young and its reverse inequalities for scalars are given. Then, several Operator and norm versions according to these inequalities are established.

The main purpose of this paper is to study a general norm on extension of a Hilbert’s type linear Operator in the continuous and discrete form. In addition to expressing the norm of a Hilbert’s type linear Operator T: L2 (0, ¥) ® L2 (0, ¥), a more general case with l > 0, for the continuous form has been studied. By putting l = 1 a norm of extension of Hilbert’s integral linear Operator is obtained. Similar results have been expressed for series when 0 < l £ 2.

In this paper, we determine the upper and lower bounds for the norm of lower triangular matrix Operators on Cesà, ro weighted (p, v)−, fractional difference sequence spaces of modulus functions. We consider the matrix Operators acting between ℓ, p(w) and Cp(v, ω, , Δ, (η, , ℓ, ), F) and identify their bounds and vice-versa. We also investigate the same characteristics for Nö, rlund and weighted mean matrix Operators.

Let $H(\mathbb{D})$ be the space of all analytic functions on the open unit disc $\mathbb{D}$ in the complex plane $\mathbb{C}$. In this paper, we investigate the boundedness and compactness of the generalized integration Operator$$I_{g,\varphi}^{(n)}(f)(z)=\int_0^z f^{(n)}(\varphi(\xi))g(\xi)\ d\xi,\quad z\in\mathbb{D},$$ from Zygmund space into weighted Dirichlet type space, where $\varphi$ is an analytic self-map of $\mathbb{D}$, $n\in\mathbb{N}$ and $g\in H(\mathbb{D})$. Also we give an estimate for the essential norm of the above Operator.

By taking into account that the computation of the numerical radius is an optimization problem, we prove, in this paper, several refinements of the numerical radius inequalities for Hilbert space Operators. It is shown, among other inequalities, that if A is a bounded linear Operator on a complex Hilbert space, then ω,(A) ≤,1 2 r |A|2 + |A∗, |2 + ∥, |A| |A∗, | + |A∗, | |A|∥, , where ω,(A), ∥, A∥, , and |A| are the numerical radius, the usual Operator norm, and the absolute value of A, respectively. This inequality provides a refinement of an earlier numerical radius inequality due to Kittaneh, namely, ω,(A) ≤,1 2 ,∥, A∥,+ A2 12 , . Some related inequalities are also discussed.

We establish an Operator extension of the following generalization of Bohr’s inequality, due to M.P. Vasi´c and D.J. Kečkić: ½Sn i=1 zi½r£ (Sni=1 a1i (1-r))r-1 Sni=1 ai ½zi½r (r<1, ziÎC, ai>0.1£i£n).We also present some inequalities related to our noncommutative generalization of Bohr’s inequality.

The Schwarz inequality and Jensen’s one are fundamental in a Hilbert space. Regarding a sesquilinear map B(X, Y ) = Y*X as an Operatorvalued inner product, we discuss Operator versions for the above inequalities and give simple conditions that the equalities hold.