Integral inequalities concerning polynomials with polar derivatives

Let p(z) be a polynomial of degree n and for any real or complex number α, Dα p(z) denotes the polar derivative of p(z) with respect to α, then D α p(z) = np(z) + (α −z)p’(z). In this paper, we consider the more general class of polynomials p(z) = a 0 + ∑ a ν z ν , 1 ≤ µ ≤ n, not vanishing in |z| 0, to estimate max |D α p(z)| in terms of max |p(z)| by involving some coefficients of p(z), where 0 < r ≤ ρ ≤ k. Interestingly, the results improve and extend other well known inequalities to polar derivative. Moreover, our results give several interesting results as special cases.

Let p(z) be a polynomial of degree n and for α ∈ C, let Dαp(z):= np(z) + (α − z)p′(z) denote the polar derivative of the polynomial p(z) with respect to α. In this paper, we obtain certain integral inequalities concerning the polar derivative of a polynomial, which besides yielding some interesting results, also includes some well-known theorems as special cases. Moreover we refine some Zygmund type inequalities for the polar derivative of a polynomial and present compact generalizations of some prior results.

Let P(z) be a polynomial of degree n and for a complex number , let DP(z) = nP(z) + (alpha − z) P′(z) denote the polar derivative of the polynomial P(z) with respect to alpha . In this paper we establish some Lr inequalities for the polar derivative of a polynomialwith restricted zeros. Our results not only generalizes some known polynomial inequalities, but also a variety of interesting results can be deduced from these by a fairly uniform procedure.

This article deals with two fundamental topics in mathematical analysis: the formulation of integral expressions and the derivation of integral inequalities. In particular, it introduces new one-parameter integral formulas and inequalities of the logarithmic type, where the integrands involve the logarithmic function in one way or another. Among the results are weighted Hölder-type integral inequalities and two different forms of Hardy-Hilbert-type integral inequalities. These results are illustrated by various examples and accompanied by rigorous proofs.

In $[4,7]$, Enestrom Kakeya theorem has stated as the following. If $f(z)=\sum_{j=0}^n k_j z^j$ is the $n^{t h}$ degree polynomial with real coefficients such that $0&lt;k_0 \leq k_1 \leq \ldots \leq k_{n-2} \leq k_{n-1} \leq k_n$ then all zeros of $\mathrm{f}(\mathrm{z})$ lies in $|z| \leq 1$. In [1], Aziz and Mahammad, showed that zeros of $f(z)$ satisfies $|z| \geq \frac{n}{n+1}$ are simple, under the same conditions. In this paper, we extend the above result to the polar derivative by relaxing the hypothesis in different ways.

In both theoretical and applied mathematics fields, integral inequalities play a critical role. Due to the behavior of the definition of convexity, both concepts convexity and integral inequality depend on each other. Therefore, the relationship between convexity and integral inequality is strong. By the importance of these concepts, we have introduced the new class of generalized convex function is known as LR‐ ‐convex interval‐valued function (LR‐ ‐convex‐IVF) by means of pseudo‐order relation (≤p). This order relation is defined on interval space. Under the new concept, first, both discrete and continuous new versions of Jensen‐type inequalities are presented by means of pseudo‐order relation. Second, several new Hermite–Hadamard (HH)‐ and Hermite–Hadamard–Fejér (HH‐Fejér)‐type inequalities are also derived for LR‐ ‐convex‐IVFs. Moreover, we have shown that our results include a wide class of new and known inequalities for LR‐ ‐convex‐IVFs and their variant forms as special cases. Useful examples that verify the applicability of the theory developed in this study are presented. It the end, we have proved that the set inclusion “⊆” coincident to pseudo‐order relation “≤p.” The concepts and techniques of this paper may be the starting point for further research in this area and used as a tool to investigate the research of probability and optimization, among others.

This paper proposes a novel generalized integral inequality based on free matrices and applies it to stability analysis of time-varying delay systems. The proposed integral inequality estimates the upper bound of the augmented quadratic term of the state and its derivative term by utilizing not only the single integral term but also the higher-order multiple integral terms. The proposed integral inequality includes several well-known integral inequalities as special cases. For the stability analysis of time-varying delay systems, a new Lyapunov-Krasovskii functional is constructed by including the double integral term with the augmented vector of the state and its derivative to utilize the proposed integral inequality when estimating the derivative of the Lyapunov-Krasovskii functional. Furthermore, to fully exploit the information on the time-varying delay, this paper divides the interval of the double integral term into two parts. Two numerical examples show that the results of the proposed method outperform those of the existing methods.

Bell (2012) catalogued unit root factors contained in linear filters used in seasonal adjustment (model-based or from the X-11 method) but noted that, for model-based seasonal adjustment, special cases could arise where filters could contain more unit root factors than was indicated by the general results. This article reviews some special cases that occur with canonical ARIMA model based adjustment in which, with some commonly used ARIMA models, the symmetric seasonal filters contain two extra nonseasonal differences (i.e., they include an extra (1 - B)(1 - F)). This increases by two the degree of polynomials in time that are annihilated by the seasonal filter and reproduced by the seasonal adjustment filter. Other results for canonical ARIMA adjustment that are reported in Bell (2012), including properties of the trend and irregular filters, and properties of the asymmetric and finite filters, are unaltered in these special cases. Special cases for seasonal adjustment with structural ARIMA component models are also briefly discussed.

The new framework of random polynomials developed by R. Pemantle, I. Rivin and the late O. Schramm has been studied in this thesis. The strong Pemantle-Rivin conjecture asks whether for random polynomials with independent and identically distributed zeros with a common probability distribution μon the complex plane, the empirical measures of their critical points would converge weakly to μ almost surely. This convergence question has connection with geometry of polynomials. S. D. Subramanian confirmed the conjecture whenμis a non-uniform distribution supported in the unit circle ∂D.&#13;\n&#13;\nIn this thesis, the conjecture has been extended to considering not only the critical points of the random polynomials, but also the zeros of their higher order, polar and Sz.-Nagy's generalized derivatives. The case thatμ(uniform or not) is supported in ∂D has been studied, where the derivative of each order has been proved to satisfy the conjecture. Subramanian's work has thereby been completed plus generalization. The same almost sure weak convergence has also been shown for polar and Sz.-Nagy's generalized derivatives, under some mild conditions. In particular, the result on polar derivative is the crux of filling up Subramanian's missing case of uniform μ.&#13;\n&#13;\nMeanwhile, the original Pemantle{Rivin conjecture asks about convergence in probability instead of the aforesaid stronger almost sure convergence. Z. Kabluchko fully solved this conjecture. In this thesis, his methodologies have been adapted to prove the analogous conjecture for random finite Blaschke products.&#13;\n&#13;\nMore precisely, for random finite Blaschke products with independent and identically distributed zeros with a common probability distributionμon the unit disc, the empirical measures of their critical points have been shown to converge weakly toμin probability. Consequently, this work contributes the very first probabilistic result to T. W. Ng and C. Y. Tsang's polynomial-finite-Blaschke-product dictionary.&#13;\n&#13;\nWhile the above works address probabilistic problems about zero distribution of polynomials and finite Blaschke products (regarded as non-euclidean polynomials by J. L. Walsh), the other part of this thesis is an application of the BKK theory in deterministic polynomial systems.&#13;\n&#13;\nThe numbers of various configurations for vortex dynamics in an ideal plane fluid without background flow had been investigated by many researchers for decades. In this thesis, fixed equilibria have been studied in the presence of a steady, incompressible and irrotational background flow. A more physically significant definition of a fixed equilibrium configuration has been suggested. Under this new definition, an attainable generic upper bound for their number has been found (in terms of the degree of the non-constant polynomial that determines the background flow and of the sizes of vortex species).&#13;\n&#13;\nOur result is established by transforming the rational function system arisen from fixed equilibria into a polynomial system, whose form is good enough to apply the aforesaid BKK theory (named after D. N. Bernshtein, A. G. Khovanskii and A. G. Kushnirenko) to show the finiteness of its number of solutions. Having this finiteness, the required bound follows from Bézout's theorem.

Let P ( X ) P(X) be a polynomial of degree N with complex coefficients and d 1 , d 2 {d_1},{d_2} two complex numbers with real part greater then − 1 -1 . Consider the Dirichlet series associated with the triple ( P ( X ) , d 1 , d 2 ) (P(X),{d_1},{d_2}) \[ L ( s ) = ∑ n = 1 ∞ P ( n ) ( n + d 1 ) s ( n + d 2 ) s . L(s) = \sum \limits _{n = 1}^\infty {\frac {{P(n)}}{{{{(n + {d_1})}^s}{{(n + {d_2})}^s}}}.} \] In this paper we get an explicit formula for L ( s ) L(s) in terms of special functions which gives meromorphic continuation of L ( s ) L(s) with at most simple poles at s = ( N + 1 − k ) / 2 , k = 0 , 1 , … s = (N + 1 - k)/2,k = 0,1, \ldots Finally we apply our explicit formula to Minakshisundaram’s zeta function of the three-dimensional sphere.

We prove the right Lax-type inequality on subarcs of the unit circle of the complex plane for complex algebraic polynomials of degree n having no zeros in the open unit disk. This is done by establishing the right Bernstein–Szegő–Videnskii type inequality for real trigonometric polynomials of degree at most n on intervals shorter than the period. The paper is closely related to recent work by B. Nagy and V. Totik. In fact, their asymptotically sharp Bernstein-type inequality for complex algebraic polynomials of degree at most n on subarcs of the unit circle is recaptured by using more elementary methods. Our discussion offers a somewhat new way to see V.S. Videnskii’s Bernstein and Markov type inequalities for trigonometric polynomials of degree at most n on intervals shorter than a period, two classical polynomial inequalities first published in 1960. A new Riesz–Schur type inequality for trigonometric polynomials is also established. Combining this with Videnskii’s Bernstein-type inequality gives Videnskii’s Markov-type inequality immediately.

The authors have proved an identity for a generalized integral operator via differentiable function. By applying the established identity and using the new and improved form of Holder’s integral inequality called Holder–Iscan integral inequality, some new generalized trapezoidal type integral inequalities have been obtain. Various special cases have been identified. Some applications to special means and error estimations of presented results have been analyzed.

New integral Hardy-type inequalities for compactly supported functions in arbitrary plane domains of hyperbolic type are given. Special cases for domains with uniformly perfect boundaries are studied and some applications are considered. The proofs depend substantially on fundamental equations and formulae for the hyperbolic metric, and use is also made of characteristics of domains in terms of moduli. These date back to Teichmüller. Bibliography: 20 titles.

On estimates for the coefficients of a polynomial

<abstract> In this paper, we establish some new Hermite-Hadamard type inequalities for differential exponential type convex functions and discuss several special cases. Moreover, in order to give the efficient of our main results, some applications for special means and error estimations are obtain. </abstract>