• Title/Summary/Keyword: q-derivative

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NEW SUBCLASS OF BI-UNIVALENT FUNCTIONS BY (p, q)-DERIVATIVE OPERATOR

  • Motamednezhad, Ahmad;Salehian, Safa
    • Honam Mathematical Journal
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    • v.41 no.2
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    • pp.381-390
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    • 2019
  • In this paper, we introduce interesting subclasses ${\mathcal{H}}^{p,q,{\beta},{\alpha}}_{{\sigma}B}$ and ${\mathcal{H}}^{p,q,{\beta}}_{{\sigma}B}({\gamma})$ of bi-univalent functions by (p, q)-derivative operator. Furthermore, we find upper bounds for the second and third coefficients for functions in these subclasses. The results presented in this paper would generalize and improve some recent works of several earlier authors.

On a q-Extension of the Leibniz Rule via Weyl Type of q-Derivative Operator

  • Purohit, Sunil Dutt
    • Kyungpook Mathematical Journal
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    • v.50 no.4
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    • pp.473-482
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    • 2010
  • In the present paper we define a q-extension of the Leibniz rule for q-derivatives via Weyl type q-derivative operator. Expansions and summation formulae for the generalized basic hypergeometric functions of one and more variables are deduced as the applications of the main result.

A RESERCH ON NONLINEAR (p, q)-DIFFERENCE EQUATION TRANSFORMABLE TO LINEAR EQUATIONS USING (p, q)-DERIVATIVE

  • ROH, KUM-HWAN;LEE, HUI YOUNG;KIM, YOUNG ROK;KANG, JUNG YOOG
    • Journal of applied mathematics & informatics
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    • v.36 no.3_4
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    • pp.271-283
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    • 2018
  • In this paper, we introduce various first order (p, q)-difference equations. We investigate solutions to equations which are linear (p, q)-difference equations and nonlinear (p, q)-difference equations. We also find some properties of (p, q)-calculus, exponential functions, and inverse function.

CERTAIN SUBCLASS OF BI-UNIVALENT FUNCTIONS ASSOCIATED WITH SYMMETRIC q-DERIVATIVE OPERATOR

  • Jae Ho Choi
    • Nonlinear Functional Analysis and Applications
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    • v.28 no.3
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    • pp.647-657
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    • 2023
  • The aim of this paper is to study certain subclass ${\tilde{S^q_{\Sigma}}}({\lambda},\,{\alpha},\,t,\,s,\,p,\,b)$ of analytic and bi-univalent functions which are defined by using symmetric q-derivative operator. We estimate the second and third coefficients of the Taylor-Maclaurin series expansions belonging to the subclass and upper bounds for Feketo-Szegö inequality. Furthermore, some relevant connections of certain special cases of the main results with those in several earlier works are also pointed out.

(p, q)-LAPLACE TRANSFORM

  • KIM, YOUNG ROK;RYOO, CHEON SEOUNG
    • Journal of applied mathematics & informatics
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    • v.36 no.5_6
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    • pp.505-519
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    • 2018
  • In this paper we define a (p, q)-Laplace transform. By using this definition, we obtain many properties including the linearity, scaling, translation, transform of derivatives, derivative of transforms, transform of integrals and so on. Finally, we solve the differential equation using the (p, q)-Laplace transform.

FEKETE-SZEGÖ INEQUALITIES FOR A NEW GENERAL SUBCLASS OF ANALYTIC FUNCTIONS INVOLVING THE (p, q)-DERIVATIVE OPERATOR

  • Bulut, Serap
    • Communications of the Korean Mathematical Society
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    • v.37 no.3
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    • pp.723-734
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    • 2022
  • In this work, we introduce a new subclass of analytic functions of complex order involving the (p, q)-derivative operator defined in the open unit disc. For this class, several Fekete-Szegö type coefficient inequalities are derived. We obtain the results of Srivastava et al. [22] as consequences of the main theorem in this study.

A RESULT ON A CONJECTURE OF W. LÜ, Q. LI AND C. YANG

  • Majumder, Sujoy
    • Bulletin of the Korean Mathematical Society
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    • v.53 no.2
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    • pp.411-421
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    • 2016
  • In this paper, we investigate the problem of transcendental entire functions that share two values with one of their derivative. Let f be a transcendental entire function, n and k be two positive integers. If $f^n-Q_1$ and $(f^n)^{(k)}-Q_2$ share 0 CM, and $n{\geq}k+1$, then $(f^n)^{(k)}{\equiv}{\frac{Q_2}{Q_1}}f^n$. Furthermore, if $Q_1=Q_2$, then $f=ce^{\frac{\lambda}{n}z}$, where $Q_1$, $Q_2$ are polynomials with $Q_1Q_2{\not\equiv}0$, and c, ${\lambda}$ are non-zero constants such that ${\lambda}^k=1$. This result shows that the Conjecture given by W. $L{\ddot{u}}$, Q. Li and C. Yang [On the transcendental entire solutions of a class of differential equations, Bull. Korean Math. Soc. 51 (2014), no. 5, 1281-1289.] is true. Also we exhibit some examples to show that the conditions of our result are the best possible.

FINITE LOGARITHMIC ORDER SOLUTIONS OF LINEAR q-DIFFERENCE EQUATIONS

  • Wen, Zhi-Tao
    • Bulletin of the Korean Mathematical Society
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    • v.51 no.1
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    • pp.83-98
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    • 2014
  • During the last decade, several papers have focused on linear q-difference equations of the form ${\sum}^n_{j=0}a_j(z)f(q^jz)=a_{n+1}(z)$ with entire or meromorphic coefficients. A tool for studying these equations is a q-difference analogue of the lemma on the logarithmic derivative, valid for meromorphic functions of finite logarithmic order ${\rho}_{log}$. It is shown, under certain assumptions, that ${\rho}_{log}(f)$ = max${{\rho}_{log}(a_j)}$ + 1. Moreover, it is illustrated that a q-Casorati determinant plays a similar role in the theory of linear q-difference equations as a Wronskian determinant in the theory of linear differential equations. As a consequence of the main results, it follows that the q-gamma function and the q-exponential functions all have logarithmic order two.

HORADAM POLYNOMIALS FOR A NEW SUBCLASS OF SAKAGUCHI-TYPE BI-UNIVALENT FUNCTIONS DEFINED BY (p, q)-DERIVATIVE OPERATOR

  • Vanithakumari Balasubramaniam;Saravanan Gunasekar;Baskaran Sudharsanan;Sibel Yalcin
    • Communications of the Korean Mathematical Society
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    • v.39 no.2
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    • pp.461-470
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    • 2024
  • In this paper, a new subclass, 𝒮𝒞𝜇,p,q𝜎 (r, s; x), of Sakaguchitype analytic bi-univalent functions defined by (p, q)-derivative operator using Horadam polynomials is constructed and investigated. The initial coefficient bounds for |a2| and |a3| are obtained. Fekete-Szegö inequalities for the class are found. Finally, we give some corollaries.