Related papers: On certain new means generated by generalized trig…
In this paper we study the convexity and concavity properties of generalized trigonometric and hyperbolic functions in case of Logarithmic mean.
In this paper, the versions of trigonometric functions of certain known inequalities for hyperbolic ones are proved, and then corresponding inequalities for means are presented.
Generalized trigonometric functions and generalized hyperbolic functions can be converted to each other by the duality formulas previously discovered by the authors. In this paper, we apply the duality formulas to prove dual pairs of…
We study the power mean inequality of the generalized trigonometric and hyperbolic functions with two parameters. The generalized $p$-trigonometric and $(p, q)$-trigonometric functions were introduced by P. Lindqvist and S. Takeuchi,…
It is shown that generalized trigonometric functions and generalized hyperbolic functions can be transformed from each other. As an application of this transformation, a number of properties for one immediately lead to the corresponding…
The main aim of this note, which can be viewed as a certain addendum to the paper \cite{2019}, is to propose several generalized inequalities for the ratio functions of trigonometric and hyperbolic functions. We basically follow the…
The generalized trigonometric functions occur as an eigenfunction of the Dirichlet problem for the one-dimensional $p-$Laplacian. The generalized hyperbolic functions are defined similarly. Some classical inequalities for trigonometric and…
We develop a new point of view to introduce families of functions, which can be identified as generalization of the ordinary trigonometric or hyperbolic functions. They are defined using a procedure based on umbral methods, inspired to the…
In this paper we shall consider some famous means such as arithmetic, harmonic, geometric, logarithmic means, etc. Inequalities involving logarithmic mean with differences among other means are presented
Investigation of the generalized trigonometric and hyperbolic functions containing two parameters has been a very active research area over the last decade. We believe, however, that their monotonicity and convexity properties with respect…
One-parameter generalizations of the logarithmic and exponential functions have been obtained as well as algebraic operators to retrieve extensivity. Analytical expressions for the successive applications of the sum or product operators on…
We study the convexity/concavity properties of the generalized $p$-trigonometric functions in the sense of P. Lindqvist with respect to the power means.
The complete elliptic integrals are generalized by using the generalized trigonometric functions with two parameters. It is shown that a particular relation holds for the generalized integrals. Moreover, as an application of the integrals,…
The generalized $p$-trigonometric and ($p,q$)-trigonometric functions were introduced by P. Lindqvist and S. Takeuchi, respectively. We prove some inequalities and present a few conjectures for the ($p,q$)-functions.
Extending the work of Freese, we further develop the theory of generalized trigonometric functions. In particular, we study to what extent the notion of polar form for the complex numbers may be generalized to arbitrary associative…
Generalised definitions of exponential, trigonometric sine and cosine and hyperbolic sine and cosine functions are given. In the lowest order, these functions correspond to ordinary exponential, trigonometric sine etc. Some of the…
Various miscellaneous functional inequalities are deduced for the so-called generalized inverse trigonometric and hyperbolic functions. For instance, functional inequalities for sums, difference and quotient of generalized inverse…
We offer new proofs, refinements as well as new results related to classical means of two variables, including the identric and logarithmic means.
Spherical means are well-known useful tool in the theory of partial differential equations with applications to solving hyperbolic and ultrahyperbolic equations and problems of integral geometry, tomography and Radon transforms. We…
In this paper we study the problem of extending means to means of higher order. We show how higher order means can be inductively defined and established in general metric spaces, in particular, in convex metric spaces. As a particular…