Integral inequalities concerning polynomials with polar derivatives

2016 ◽  
Vol 25 (1) ◽  
pp. 77-84
Author(s):  
ABDULLAH MIR ◽  
◽  
SHAHISTA BASHIR ◽  
Keyword(s):  
Complex Number ◽  
Integral Inequality ◽  
Integral Inequalities ◽  
Polar Derivative ◽  
Special Cases ◽  
Polar Derivatives ◽  
Derivative Of A Polynomial

Let P(z) be a polynomial of degree n and for any complex number α, let DαP(z) = nP(z) + (α − z)P 0 (z) denote the polar derivative of P(z) with respect to a complex number α. In this paper, we present an integral inequality for the polar derivative of a polynomial P(z). Our result includes as special cases several interesting generalizations of some Zygmund type inequalities for polynomials.

scholarly journals Some integral inequalities for the polar derivative of polynomials

2019 ◽  
Vol 106 (120) ◽  
pp. 85-94 ◽  
Author(s):  
Prasanna Kumar
Keyword(s):  
Integral Inequalities ◽  
Polar Derivative ◽  
Special Cases ◽  
Polar Derivatives

As a generalization of well-known result due to Turan [24] for polynomials having all their zeros in |z| ? 1, Malik [17] proved that, if P(z) is a polynomial of degree n, having all its zeros in |z| ? 1, then for any ? > 0, n{?2?0|P(ei?)|?d?}1/? ? {?2?0|1+ei?|?d?}1/? max |z|=1 |P?(z)|. We generalize the above inequality to polar derivatives, which as special cases include several known results in this area. Besides the paper contains some more results that generalize and sharpen several results known in this direction.

scholarly journals A new generalization of some quantum integral inequalities for quantum differentiable convex functions

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Yi-Xia Li ◽  
Muhammad Aamir Ali ◽  
Hüseyin Budak ◽  
Mujahid Abbas ◽  
Yu-Ming Chu
Keyword(s):  
Integral Inequality ◽  
Convex Functions ◽  
Integral Identity ◽  
Integral Inequalities ◽  
Quantum Integral ◽  
Special Cases

AbstractIn this paper, we offer a new quantum integral identity, the result is then used to obtain some new estimates of Hermite–Hadamard inequalities for quantum integrals. The results presented in this paper are generalizations of the comparable results in the literature on Hermite–Hadamard inequalities. Several inequalities, such as the midpoint-like integral inequality, the Simpson-like integral inequality, the averaged midpoint–trapezoid-like integral inequality, and the trapezoid-like integral inequality, are obtained as special cases of our main results.

scholarly journals Some integral inequalities for approximately h—convex functions and their applications

2021 ◽  
Vol 40 (2) ◽  
pp. 481-504
Author(s):  
Artion Kashuri ◽  
Muhammad Raees ◽  
Matloob Anwar
Keyword(s):  
Convex Function ◽  
Integral Inequality ◽  
Convex Functions ◽  
Integral Inequalities ◽  
Integral Type ◽  
Special Cases ◽  
Hölder’S Integral Inequality ◽  
Error Estimations

In this paper, by applying the new and improved form of Hölder’s integral inequality called Hölder—Íşcan integral inequality three inequalities of Hermite—Hadamard and Hadamard integral type for (h, d)—convex functions have been established. Various special cases including classes for instance, h—convex, s—convex function of Breckner and Godunova—Levin—Dragomir and strong versions of the aforementioned types of convex functions have been identified. Some applications to error estimations of presented results have been analyzed. At the end, a briefly conclusion is given.

scholarly journals General Raina fractional integral inequalities on coordinates of convex functions

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Dumitru Baleanu ◽  
Artion Kashuri ◽  
Pshtiwan Othman Mohammed ◽  
Badreddine Meftah
Keyword(s):  
Mathematical Model ◽  
Fractional Calculus ◽  
Convex Function ◽  
Mathematical Analysis ◽  
Integral Inequality ◽  
Convex Functions ◽  
Fractional Integral ◽  
Integral Inequalities ◽  
Differentiable Functions ◽  
Special Cases

AbstractIntegral inequality is an interesting mathematical model due to its wide and significant applications in mathematical analysis and fractional calculus. In this study, authors have established some generalized Raina fractional integral inequalities using an $(l_{1},h_{1})$ ( l 1 , h 1 ) -$(l_{2},h_{2})$ ( l 2 , h 2 ) -convex function on coordinates. Also, we obtain an integral identity for partial differentiable functions. As an effect of this result, two interesting integral inequalities for the $(l_{1},h_{1})$ ( l 1 , h 1 ) -$(l_{2},h_{2})$ ( l 2 , h 2 ) -convex function on coordinates are given. Finally, we can say that our findings recapture some recent results as special cases.

scholarly journals Generalizations of some Zygmund-type integral inequalities for polar derivatives of a complex polynomial

2021 ◽  
Vol 110 (124) ◽  
pp. 57-69
Author(s):  
Abdullah Mir
Keyword(s):  
Unit Disk ◽  
Complex Plane ◽  
Integral Inequalities ◽  
Complex Polynomial ◽  
Algebraic Polynomials ◽  
Polar Derivative ◽  
Type Integral ◽  
Polar Derivatives ◽  
Derivatives Of

We prove some results for algebraic polynomials in the complex plane that relate the L-norm of the polar derivative of a complex polynomial and the polynomial under some conditions. The obtained results include several interesting generalizations of some Zygmund-type integral inequalities for polynomials and derive polar derivative analogues of some classical Bernsteintype inequalities for the sup-norms on the unit disk as well.

Integral inequalities for polar derivative of a polynomial

2021 ◽  
Author(s):  
Maisnam Triveni Devi ◽  
Kshetrimayum Krishnadas ◽  
N. Reingachan ◽  
Barchand Chanam
Keyword(s):  
Integral Inequalities ◽  
Polar Derivative ◽  
Derivative Of A Polynomial

scholarly journals Fractional Integral Inequalities via Hadamard’s Fractional Integral

2014 ◽  
Vol 2014 ◽  
pp. 1-11 ◽  
Author(s):  
Weerawat Sudsutad ◽  
Sotiris K. Ntouyas ◽  
Jessada Tariboon
Keyword(s):  
Integral Inequality ◽  
Fractional Integral ◽  
Integral Inequalities ◽  
Special Cases ◽  
Hadamard Fractional Integral

We establish new fractional integral inequalities, via Hadamard’s fractional integral. Several new integral inequalities are obtained, including a Grüss type Hadamard fractional integral inequality, by using Young and weighted AM-GM inequalities. Many special cases are also discussed.

scholarly journals Some Bounds for the Polar Derivative of a Polynomial

2018 ◽  
Vol 2018 ◽  
pp. 1-4
Author(s):  
Jiraphorn Somsuwan ◽  
Keaitsuda Maneeruk Nakprasit
Keyword(s):  
Lower Bound ◽  
Complex Number ◽  
Upper Bound ◽  
Polar Derivative ◽  
Derivative Of A Polynomial

The polar derivative of a polynomial p(z) of degree n with respect to a complex number α is a polynomial np(z)+α-zp′(z), denoted by Dαp(z). Let 1≤R≤k. For a polynomial p(z) of degree n having all its zeros in z≤k, we investigate a lower bound of modulus of Dαp(z) on z=R. Furthermore, we present an upper bound of modulus of Dαp(z) on z=R for a polynomial p(z) of degree n having no zero in z<k. In particular, our results in case R=1 generalize some well-known inequalities.

scholarly journals Fractional Integral Inequalities for Exponentially Nonconvex Functions and Their Applications

2021 ◽  
Vol 5 (3) ◽  
pp. 80
Author(s):  
Hari Mohan Srivastava ◽  
Artion Kashuri ◽  
Pshtiwan Othman Mohammed ◽  
Dumitru Baleanu ◽  
Y. S. Hamed
Keyword(s):  
Integral Inequalities ◽  
Fractional Integrals ◽  
Auxiliary Result ◽  
Wright Function ◽  
Nonconvex Functions ◽  
Special Cases ◽  
Type Integral ◽  
Generic Class ◽  
Two Parameters ◽  
Class Of Functions

In this paper, the authors define a new generic class of functions involving a certain modified Fox–Wright function. A useful identity using fractional integrals and this modified Fox–Wright function with two parameters is also found. Applying this as an auxiliary result, we establish some Hermite–Hadamard-type integral inequalities by using the above-mentioned class of functions. Some special cases are derived with relevant details. Moreover, in order to show the efficiency of our main results, an application for error estimation is obtained as well.

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