Cusa-Huygens, Wilker and Huygens Type Inequalities for Generalized Hyperbolic Functions

In this paper, we establish Cusa-Huygens, Wilker and Huygens type inequalities for certain generalizations of the hyperbolic functions. From the established results, we recover some previous results as particular cases.

Also, in a recent work, the Huygens-type inequality was established among other things by Bagul and Chesneau [1].
Motivated by the results (2), (5), (6), (8) and (9), the objective of this paper is to establish analogous inequalities concerning certain generalizations of the hyperbolic functions. The established results serve as generalizations of the previous results.

Preliminary Definitions
In a bid to generalize a previous work [9], the authors of [10] gave the following generalizations of the hyperbolic functions.
Definition 2.1. The generalized hyperbolic cosine, hyperbolic sine and hyperbolic tangent functions are respectively defined as [10] cosh where a > 1 and z ∈ R.
These generalized functions satisfy the following identities.
As pointed out in [10], several other identities can be derived from (10), (11) and (12). When a = e, where e = 2.71828... is the Euler's number, then the above definitions and identities reduce to their ordinary counterparts.

Results and Discussion
Lemma 3.1. The inequality holds for z ∈ R \ {0}.
Since the function sinha(z) z is increasing for z > 0 and decreasing for z < 0, then Lemma 3.1 implies the following generalized result.
Proof. Since the functions in each term of the inequality are even, it suffices to prove the case for z > 0. Let z > 0 and h be defined as hold for z ∈ R \ {0}.
Proof. Let z ∈ R \ {0}. Then by the AM-GM inequality and Lemma 3.1, we obtain which gives (31). To prove (32), it suffices to prove the case for z > 0. Let z > 0 and define ψ(z) by where 1 < a ≤ e. Then by differentiating, applying the AM-GM inequality and Lemma 3.1, we obtain . cosh a (z) Thus ψ(z) is increasing. Hence which gives (32).
Proof. It suffices to prove the case for z > 0. Let z > 0 and let h be defined as Then Remark 3.13. When a = e, then inequality (35) reduces to (8).
Proof. It suffices to prove the case for z > 0. Let z > 0 and δ be defined as .  > 0, since sinh a (z) > (ln a)z and cosh a (z) > 1 . Hence g (z) is increasing and so, g (z) > g (0) = 0. Thus, g(z) is increasing and so g(z) > g(0) = 0. This yields the right-hand side of (39) and that completes the proof.