Using bounds for the function to be optimized plus score of current best solution enables B&B to search parts of the candidate space only.
Very recently, Lv, Yang, Luo, and Zheng [19] gave a new type of bounds for the function (exp (xcot x-1)).
The importance of this definition is that the upper and lower solutions give upper and lower bounds for the function (rho(t)) for (tgeq0).
In [2], motivated by the relation g ( x ) = 2 g ( 1 - 1 - x ) with g as in (2.9), the authors asked the question about the bounds for the function S t) defined by S ( t ) : = g ( t ) g ( 1 - 1 - t ), t ∈ ( 0, 1 ), where g(t):= tF a, b; a + b; t), a, b > 0. An answer follows instantly applying Corollary 3.8.
In the article, we present new bounds for the function (e^{tcot(t -1}) on t -1}nterval ((0, pi/2)) and find sharp estimations for the Sintervalgral and the Catalan constant based on a new monotonicity criterion for the quotient of power series, which refine the Redheffer and Becker-Stark type inequalities for tangent function.
Unfortunately, in the article we cannot deal with the monotonicity for the function (R x)) defined by (1.5) and present the bounds for the function (I_{p}(x)) given by (1.2) in the case of (pin 0,1)); we leave it as an open problem to the reader.
