History has no place in rebellion's logic; instead, rebels have "uncompromising" means defined by their axiom, which always governs their actions.
We begin with the form of hyperbolic functions of Stolarsky means defined by (1.1) to introduce the two-parameter sine functions.
The monotonicities and log-convexities of S h p, q ( b, a ) and of C h p, q ( b, a ) in parameters p and q are clearly the same as those of S h ( p, q, t ) and of C h ( p, q, t ), which are in turn equivalent to those of Stolarsky means defined by (1.1) and of Gini means defined by (1.2), respectively.
In the same way, the Gini means defined by (1.2) can be expressed in hyperbolic functions by letting t = ln b / a : G p, q ( a, b ) = a b C h ( p, q, t ), where C h ( p, q, t ) = { ( cosh p t cosh q t ) 1 / ( p − q ), p ≠ q, e t tanh p t, p = q ≠ 0, 1, p = q = 0. (4.1).
In the case of the car competition, contestants must enter vehicles that are safe to drive and affordable to build, through means defined by a dense set of rules.
Now, we deduce the monotonicity of means defined by (38) in the form of Dresher's inequality as follows.
are means of a and b, where S B ( a, b ) is the Schwab-Borchardt mean defined by (1.3).
are means of a and b, where T = T ( a, b ) is the second Seiffert mean defined by (1.6).
are means of a and b, where P, T are the first and second Seiffert mean defined by (1.5) and (1.6), U is defined by (3.16).
The third class is mainly composed of trigonometric functions and their inverses, for example, the first part of Schwab-Borchardt mean defined by (1.3), the first and second Seiffert means [9, 10] defined by P = P ( a, b ) = a − b 2 arcsin a − b a + b, (1.5).
Generalized logarithmic mean defined by (2.16) is convex for and concave for.
