• The Pure and Applied Mathematics
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• v.28 no.4
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• pp.343-353
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• 2021

In this paper, we obtain some inequalities concerning the class of generalized derivative and generalized polar derivative which are analogous respectively to the ordinary derivative and polar derivative of polynomials.

• Journal of applied mathematics & informatics
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• v.42 no.4
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• pp.915-930
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• 2024

We obtain new fractional integral inequalities which generalize certain inequalities given in [16]. Generalized inequalities can be used to study global existence results for fractional differential equations.

• The Pure and Applied Mathematics
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• v.13 no.4 s.34
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• pp.261-267
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• 2006

A local convergence analysis of Newton's method for perturbed generalized equations is provided in a Banach space setting. Using center Lipschitzian conditions which are actually needed instead of Lipschitzian hypotheses on the $Fr\'{e}chet$-derivative of the operator involved and more precise estimates under less computational cost we provide a finer convergence analysis of Newton's method than before [5]-[7].

• Journal of applied mathematics & informatics
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• v.36 no.1_2
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• pp.107-114
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• 2018

We establish some new ostrowski type inequalities for MT-convex function including first order derivative via Niemann-Trouvaille fractional integral. It is interesting to mention that our results provide new estimates on these types of integral inequalities for MT-convex functions.

• Korean Journal of Mathematics
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• v.31 no.2
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• pp.161-180
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• 2023

In this article, we will utilize (α, m)-convexity to create a new form of Simpson-Newton inequalities in quantum calculus by using q_𝝔1-integral and q_𝝔1-derivative. Newly discovered inequalities can be transformed into quantum Newton and quantum Simpson for generalized convexity. Additionally, this article demonstrates how some recently created inequalities are simply the extensions of some previously existing inequalities. The main findings are generalizations of numerous results that already exist in the literature, and some fundamental inequalities, such as Hölder's and Power mean, have been used to acquire new bounds.

• East Asian mathematical journal
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• v.14 no.1
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• pp.107-116
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• 1998

This paper deals with functions of the form $f(z)=a_1z-{\sum}{\limits}_{n=2}^{\infty}a_nz^n(a_1>0,\;a_n{\geqslant}0)$ with $(1-{\mu})f(z_0)/z_0+{\mu}f'(z_0)=1(-1. We introduce the class $\varphi({\mu},{\eta},{\gamma},{\delta},A,B;\;z_0)$ with generalized fractional derivatives. Also we have obtained coefficient inequalities, distortion theorem and radious problem of functions belonging to the calss $\varphi({\mu},{\eta},{\gamma},{\delta},A,B;\;z_0)$.

• Nonlinear Functional Analysis and Applications
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• v.26 no.4
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• pp.855-868
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• 2021

If p(z) is a polynomial of degree n having all its zeros in |z| ≤ k, k ≤ 1, then for 𝜌R ≥ k² and 𝜌 ≤ R, Aziz and Zargar [4] proved that $${\max_{{\mid}z{\mid}=1}}{\mid}p^{\prime}(z){\mid}{\geq}n{\frac{(R+k)^{n-1}}{({\rho}+k)^n}}\{{\max_{{\mid}z{\mid}=1}}{\mid}p(z){\mid}+{\min_{{\mid}z{\mid}=k}}{\mid}p(z){\mid}\}$$. We prove a generalized L^r extension of the above result for a more general class of polynomials $p(z)=a_nz^n+\sum\limits_{{\nu}={\mu}}^{n}a_n-_{\nu}z^{n-\nu}$, $1{\leq}{\mu}{\leq}n$. We also obtain another L^r analogue of a result for the above general class of polynomials proved by Chanam and Dewan [6].

• Communications of the Korean Mathematical Society
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• v.38 no.4
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• pp.1111-1126
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• 2023

In this paper, we define a new subclass of k-uniformly starlike functions of order γ (0 ≤ γ < 1) by using certain generalized q-integral operator. We explore geometric interpretation of the functions in this class by connecting it with conic domains. We also investigate q-sufficient coefficient condition, q-Fekete-Szegö inequalities, q-Bieberbach-De Branges type coefficient estimates and radius problem for functions in this class. We conclude this paper by introducing an analogous subclass of k-uniformly convex functions of order γ by using the generalized q-integral operator. We omit the results for this new class because they can be directly translated from the corresponding results of our main class.

• Korean Journal of Mathematics
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• v.17 no.2
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• pp.127-145
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• 2009

In this paper we introduce a new subclass $K_{\mu}^{\lambda},{\phi},{\eta}(n;{\rho};{\alpha})$ of analytic and multivalent functions with negative coefficients using fractional calculus operators. Connections to the well known and some new subclasses are discussed. A necessary and sufficient condition for a function to be in $K_{\mu}^{\lambda},{\phi},{\eta}(n;{\rho};{\alpha})$ is obtained. Several distortion inequalities involving fractional integral and fractional derivative operators are also presented. We also give results for radius of starlikeness, convexity and close-to-convexity and inclusion property for functions in the subclass. Modified Hadamard product, application of class preserving integral operator and other interesting properties are also discussed.

• Journal of applied mathematics & informatics
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• v.33 no.3_4
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• pp.275-292
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• 2015

This paper is mainly concerned with the existence of solution for nonlinear impulsive fractional dynamic equations on a special time scale.We introduce the new concept and propositions of fractional q-integral, q-derivative, and α-Lipschitz in the paper. By using a new fixed point theorem, we obtain some new existence results of solutions via some generalized singular Gronwall inequalities on time scales. Further, an interesting example is presented to illustrate the theory.