The Wirtinger-Steklov inequality between the norm of a periodic function and the norm of the positive cutoff of its derivative

We study the sharp constant in the inequality between the L _p -mean (p ≥ 0) of a 2π-periodic function with zero mean value and the L _q -norm (q ≥ 1) of the positive cutoff of its derivative. We obtain estimates of the constant from below for 0 ≤ p ≤ ∞ and from above for 1 ≤ p ≤ ∞ for an arbitrary 1 ≤ q ≤ ∞. We write out the values of the sharp constant in the cases p = 2, 1 ≤ q ≤ ∞ and p = ∞, 1 ≤ q ≤ ∞.     

MATRIX NORM VERSIONS OF THE KANTOROVICH INEQUALITY AND ITS APPLICATIONS

In this paper we discuss the generalizations of the Kantorovich inequality and obtain some generalized Kantorovich inequalities in the sense of matrix norm. We further illustrate how to use these inequalities to determine the lower bound of relative efficiency of the parameter estimate in linear model.     

On the norm of a dc function

It is well known that, if a vector-valued function can be written as difference of componentwise convex functions, the norm of such function inherits this property. In this note we show that, if the norm in use is monotonic in the positive orthant and the functions are non-negative, a sharper decomposition can be obtained.     

An algebroid function and its derivative

In this paper, we investigate the value distribution of an algebroid function and its derivative, and obtain two inequations between Nevanlinna characteristic function of an algebroid function and that of its derivative. We extend Chuang Chitai's theorem of meromorphic functions to algebroid functions.     

A fixed point theorem and a norm inequality for operator means

There is one to one correspondence between positive operator monotone functions on (0, ∞) and operator connections. For a symmetric connection σ, it is proved that the map X → (AσX)σ(BσX) from positive operators on a Hilbert space to itself, has a unique fixed point. Here σ denotes the dual of σ. It is also proved that |||AσB||| |||A|||σ|||B||| for all unitarily invariant norms ||| · ||| and for all positive operators A,B.     

Brascamp-Lieb inequality for positive double John basis and its reverse

In this paper, we establish the Brascamp-Lieb inequality for positive double John basis and its reverse. As their applications, we estimate the upper and lower bounds for the volume product of two unit balls with the given norms. Moreover, the Loomis-Whitney inequality for positive double John basis is obtained.     

Uniqueness of entire function sharing a small function with its derivative

This paper is devoted to studying the relationship between an entire function and its derivative when they share one small function. We generalize some previous results of Gundersen and Yang [G. Gundersen, L.Z. Yang, Entire functions that share one value with one or two of their derivatives, J. Math. Anal. Appl. 223 (1998) 85–95], Chang and Zhu [J. Chang, Y. Zhu, Entire functions that share a small function with their derivatives, J. Math. Anal. Appl. 351 (2009) 491–496].     

On an inequality concerning the polar derivative of a polynomial

In this paper, we present a correct proof of an L _p -inequality concerning the polar derivative of a polynomial with restricted zeros. We also extend Zygmund’s inequality to the polar derivative of a polynomial.     

The derivative of a continuous nonconstant function

Using elementary ideas and techniques, we prove (Theorem 2) that for a nonconstant differentiable function , a < b, the set S(f) = {x ε [a, b] : f'(x) ≠ 0} cannot be negligible. This result remains valid if f fails to be differentiable on a countable subset of [a, b].     

The logarithmic derivative of a meromorphic function

A well-known lemma on the logarithmic derivative for a function f(z), f(0) = 1 (0 < r="> m( r,\fracf￠f ) < ln+ { \fracT(r,f)r\fracrr- r } + 5.8501.m\left( {r,\frac{{f'}}{f}} \right)< \ln + \left\{ {\frac{{T(\rho ,f)}}{r}\frac{\rho }{{\rho - r}}} \right\} + 5.8501.     

Extremal problem of the norm of an intermediate derivative

Translated from Matematicheskie Zametki, Vol. 49, No. 2, pp. 45–54, February, 1991.     

Improvement of an inequality of Hardy containing an estimate of the size of an intermediate derivative of a function

Translated from Matematicheskie Zametki, Vol. 50, No. 4, pp. 114–122, October, 1991.     

Entire functions that share a small function with its derivative

In this paper, by studying the properties of meromorphic functions which have few zeros and poles, we find all the entire functions f(z) which share a small and finite order meromorphic function a(z) with its derivative, and f(n)(z)−a(z)=0 whenever f(z)−a(z)=0 (n?2). This result is a generalization of several previous results.     

An inequality concerning the growth bound of an evolution family and the norm of a convolution operator

Let \(\mathcal {U}=\{U(t,s)\}_{t\ge s\ge 0}\) be a strongly continuous and exponentially bounded evolution family acting on a complex Banach space X and let \(\mathcal {X}\) be a certain Banach function space of X-valued functions. We prove that the growth bound of the family \(\mathcal {U}\) is less than or equal to \(-\frac{1}{c(\mathcal {U}, \mathcal {X})}\) provided that the convolution operator \(f\mapsto \mathcal {U}*f\) acts on \(\mathcal {X}.\) It is well known that under the latter assumption, the convolution operator is bounded and then \(c(\mathcal {U}, \mathcal {X})\) denotes (ad-hoc) its norm in \(\mathcal {L}(\mathcal {X}).\) As a consequence, we prove that if \(\sup \nolimits _{s\ge 0}\int \nolimits _{s}^\infty \Vert U(t,s)\Vert dt=u_1(\mathcal {U})<\infty ,\) then \(\omega _0(\mathcal {U})u_1(\mathcal {U})\le -1.\) Finally, we give an example showing that the accuracy of the estimates may be quite accurate.