Applications of a duality between generalized trigonometric and hyperbolic functions

Applications of a duality between generalized trigonometric and hyperbolic functions II

“…Note that if p = q, then tam p,q x = tan p,q x and tamh p,q x = tanh p,q x. In [8], we proved the following duality properties between GTFs and GHFs. This property remains important in the present paper.…”

“…In this section, we summarize the definitions and some properties of GTFs and GHFs (see [8] for more details). The relationship between GTFs and GHFs can be seen by making the range of parameters in the functions wider than the conventional definition.…”

“…However, for practical purposes, the following modified functions are more convenient than the functions above: tam p,q x := sin p,q x cos p/q p,q x , tamh p,q x := sinh p,q x cosh p/q p,q x . These modified functions were first introduced in [8,12] with the symbols τ p,q , τp,q , respectively. Note that if p = q, then tam p,q x = tan p,q x and tamh p,q x = tanh p,q x.…”

“…In this paper, following [8], we will generalize the old and vigorously studied Wilker inequalities, Huygens inequalities and (relaxed) Cusa-Huygens inequalities to GTFs and GHFs. In fact, previous works, e.g., [5,7,9,14,15], have made various generalizations of these inequalities, but the trigonometric and hyperbolic versions are not in dual pairs.…”

“…This paper is organized as follows. Section 2 summarizes the definitions of GTFs and GHFs and their properties, including the duality formulas obtained in [8]. Here, the conditions imposed on the parameters contained in these functions are more extended than usual.…”