Monotonicity properties and inequalities related to generalized Grötzsch ring functions

Abstract In the paper, the authors present some monotonicity properties and some sharp inequalities for the generalized Grötzsch ring function and related elementary functions. Consequently, the authors obtain new bounds for solutions of the Ramanujan generalized modular equation.

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Monotonicity properties and bounds involving the two-parameter generalized Grötzsch ring function

Journal of Inequalities and Applications ◽

10.1186/s13660-020-02327-7 ◽

2020 ◽

Vol 2020 (1) ◽

Cited By ~ 6

Author(s):

Guo-Jing Hai ◽

Tie-Hong Zhao

Keyword(s):

Monotonicity Properties ◽

Generalized Grötzsch Ring Function ◽

Two Parameter

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Second-order integrable Lagrangians and WDVV equations

Letters in Mathematical Physics ◽

10.1007/s11005-021-01403-3 ◽

2021 ◽

Vol 111 (2) ◽

Author(s):

E. V. Ferapontov ◽

M. V. Pavlov ◽

Lingling Xue

Keyword(s):

Lagrangian Density ◽

Theta Functions ◽

Second Order ◽

Elementary Functions ◽

Lagrange Equations ◽

Wdvv Equations ◽

Integrability Conditions ◽

Jacobi Theta Functions

AbstractWe investigate the integrability of Euler–Lagrange equations associated with 2D second-order Lagrangians of the form $$\begin{aligned} \int f(u_{xx},u_{xy},u_{yy})\ \mathrm{d}x\mathrm{d}y. \end{aligned}$$ ∫ f ( u xx , u xy , u yy ) d x d y . By deriving integrability conditions for the Lagrangian density f, examples of integrable Lagrangians expressible via elementary functions, Jacobi theta functions and dilogarithms are constructed. A link of second-order integrable Lagrangians to WDVV equations is established. Generalisations to 3D second-order integrable Lagrangians are also discussed.

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Quantum periods and TBA-like equations for a class of Calabi-Yau geometries

Journal of High Energy Physics ◽

10.1007/jhep01(2021)002 ◽

2021 ◽

Vol 2021 (1) ◽

Author(s):

Bao-ning Du ◽

Min-xin Huang

Keyword(s):

Large Class ◽

Bethe Ansatz ◽

Difference Equations ◽

The Other ◽

Elementary Functions ◽

Quantum Dilogarithm ◽

Thermodynamic Bethe Ansatz ◽

Dilogarithm Function ◽

Quantum Operators

Abstract We continue the study of a novel relation between quantum periods and TBA(Thermodynamic Bethe Ansatz)-like difference equations, generalize previous works to a large class of Calabi-Yau geometries described by three-term quantum operators. We give two methods to derive the TBA-like equations. One method uses only elementary functions while the other method uses Faddeev’s quantum dilogarithm function. The two approaches provide different realizations of TBA-like equations which are nevertheless related to the same quantum period.

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Series and Products for Elementary Functions

Series and Products in the Development of Mathematics ◽

10.1017/9781108709453.016 ◽

2021 ◽

pp. 365-395

Keyword(s):

Elementary Functions

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Application of the Efros Theorem to the Function Represented by the Inverse Laplace Transform of s−μ exp(−sν)

Symmetry ◽

10.3390/sym13020354 ◽

2021 ◽

Vol 13 (2) ◽

pp. 354

Author(s):

Alexander Apelblat ◽

Francesco Mainardi

Keyword(s):

Laplace Transform ◽

Operational Calculus ◽

Inverse Laplace Transform ◽

Elementary Functions ◽

Hyperbolic Functions ◽

Error Functions ◽

Convolution Integrals ◽

Special Case ◽

Integral Identities ◽

Finite Integrals

Using a special case of the Efros theorem which was derived by Wlodarski, and operational calculus, it was possible to derive many infinite integrals, finite integrals and integral identities for the function represented by the inverse Laplace transform. The integral identities are mainly in terms of convolution integrals with the Mittag–Leffler and Volterra functions. The integrands of determined integrals include elementary functions (power, exponential, logarithmic, trigonometric and hyperbolic functions) and the error functions, the Mittag–Leffler functions and the Volterra functions. Some properties of the inverse Laplace transform of s−μexp(−sν) with μ≥0 and 0<ν<1 are presented.

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