A proof of two conjectures of Chao-Ping Chen for inverse trigonometric functions

Abstract

Abstract. In this paper we prove two conjectures stated by Chao-Ping Chen in [Int. Trans. Spec. Funct. 23:12 (2012), 865-873], using a method for proving inequalities of mixed trigonometric polynomial functions.Mathematics subject classification (2010): 26D05.

The natural algorithmic approach of mixed trigonometric-polynomial problems

“…Our general algorithm associated with the natural approach method can be successfully applied to prove a wide category of classical MTP inequalities. For example, the Natural Approach algorithm has recently been used to prove several open problems that involve MTP inequalities (see, e.g., [ 8 – 12 ]).…”

“…It is our contention that the Natural Approach algorithm can be used to introduce and solve other new similar results. Chen [ 4 ] used a similar method to prove the following inequalities, for every : and then he proposed the following inequalities as a conjecture: and Very recently, Malešević et al [ 12 ] solved this open problem using the same procedure, i.e., the natural approach method, associated with upwards and downwards approximations of the inverse trigonometric functions.…”

“…In this case, in order to use the results of Lemmas 1 - 2 , we are forced to choose large enough values of such that . Note that a higher value of k implies a higher degree of the downward Taylor approximations and of the polynomial in ( 2 ) (for instance, see [ 10 ] and [ 12 ]).…”

A proof of an open problem of Yusuke Nishizawa for a power-exponential function

“…This paper proved an open problem stated by Nishizawa in [1], applying computation method from [4] and [14]. We note that proofs of polynomial inequalities (17), (28), (45) and (50) can be based on reducing (by differentiation) of the corresponding polynomials to polynomials of a degree up to four (as illustrated in papers [12]− [15]), which allows symbolic radical representation of roots.…”

Sharp generalized Papenfuss-Bach-type inequality

“…Theorem 1.6 can be proved using the methods and algorithms proposed in [12] and [7]. Some of the open problems have been proved by these methods in [9] and [10].…”