Ling Zhu

Sharpening Jordan's inequality and the Yang Le inequality

Abstract In this work, the following inequality: sin x x ≤ 2 π + π − 2 π 3 ( π 2 − 4 x 2 ) , x ∈ ( 0 , π / 2 ] is established. An application of this inequality gives an improvement of the Yang Le inequality [C.J. Zhao, Generalization and strengthening of the Yang Le inequality, Math. Practice Theory 30 (4) (2000) 493–497 (in Chinese)]: ( n − 1 ) ∑ k = 1 n cos 2 λ A k − 2 cos λ π ∑ 1 ≤ i j ≤ n cos λ A i cos λ A j ≤ 4 n 2 ( λ 3 + λ ( 1 − λ 2 ) 2 π ) 2 , where A i > 0 ( i = 1 , 2 , … , n ) , ∑ i = 1 n A i ≤ π , 0 ≤ λ ≤ 1 , and n ≥ 2 is a natural number.

Sharpening Jordan's inequality and Yang Le inequality, II

Abstract In this work, two refined forms of Jordan’s inequality: (a) 2 π + 1 π 3 ( π 2 − 4 x 2 ) + 12 − π 2 16 π 5 ( π 2 − 4 x 2 ) 2 ≤ sin x x ≤ 2 π + 1 π 3 ( π 2 − 4 x 2 ) + π − 3 π 5 ( π 2 − 4 x 2 ) 2 and (b) 2 π + 1 π 3 ( π 2 − 4 x 2 ) + 4 ( π − 3 ) π 3 ( x − π 2 ) 2 ≤ sin x x ≤ 2 π + 1 π 3 ( π 2 − 4 x 2 ) + 12 − π 2 π 3 ( x − π 2 ) 2 are established, where x ∈ ( 0 , π / 2 ] . The applications of the two results above give some new improvement of the Yang Le inequality.

A source of inequalities for circular functions

In this work, a basic theorem is established and it is found to be a source of inequalities for circular functions, such as Cusas, Huygens, Wilkers, the Sandor-Bencze, Carlsons, the Shafer-Fink inequality, and the one in the problem of Oppenheim. Furthermore, these inequalities described above will be extended by this basic theorem.

Six new Redheffer-type inequalities for circular and hyperbolic functions

In this paper, six new Redheffer-type inequalities involving circular functions and hyperbolic functions are established.

A general refinement of Jordan-type inequality

In this work, a general form of Jordans inequality: P2N(x)+aN+1(@p^2-4x^2)^N^+^[emailxa0protected][emailxa0protected]?P2N(x)[emailxa0protected]?n=0Na[emailxa0protected]^2^[emailxa0protected]^2^(^N^+^1^)(@p^2-4x^2)^N^+^1 is established, where [emailxa0protected]?(0,@p/2],P2N(x)[emailxa0protected]?n=0^Nan(@p^2-4x^2)^n,a[emailxa0protected],a[emailxa0protected]^3,an+1=2n+12(n+1)@p^2an-116n(n+1)@p^2an-1, and N>=0 is a natural number. The applications of the above result give the general improvement of the Yang Le inequality and a new infinite series (sinx)/[emailxa0protected]?n=0^~an(@p^2-4x^2)^n for 0<|x|@[emailxa0protected]/2.

Some new inequalities of the Huygens type

In this note, we show some new inequalities of the Huygens type for circular functions, hyperbolic functions, and the reciprocals of circular and hyperbolic functions by using a monotone form of lHospitals rule.

General forms of Jordan and Yang Le inequalities

In this work, general forms of Jordan and Yang Le inequalities are established.

Jordan type inequalities involving the Bessel and modified Bessel functions

In this work, two general forms of Jordans inequalities for the Bessel and modified Bessel functions are established, and proved by using lHospitals rule for monotonicity and some properties of the spherical Bessel and the modified spherical Bessel functions of the first kind. The applications of the results above give two new infinite series for the Bessel and modified Bessel functions.

On New Wilker-Type Inequalities

In this paper, we present and prove a new Wilker-type inequality for hyperbolic functions. We also give a simple proof of the counterpart of the above inequality for the circular functions.

Sharpening Redheffer-type inequalities for circular functions

In this note, some new sharpened Redheffer-type inequalities involving circular functions are established.