A CLASS OF NEW TRIGONOMETRIC INEQUALITIES AND THEIR SHARPENINGS

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1 Journal of Mathematical Inequalities Volume, Number 3 (008), A CLASS OF NEW TRIGONOMETRIC INEQUALITIES AND THEIR SHARPENINGS KUN ZHU, HONG ZHANG AND KAIQING WENG (communicated by V. Volenec) Abstract. Some sufficient or necessary conditions for Schur-convexity of a function of two variables F(x, y) =(f (y) f (x))/(g(y) g(x)) were considered. These results are applied to establish a class new inequalities in a triangle. In the fourth section we prove two theorems for a kind of symmetric function. These theorems are used to sharpen some of the inequalities and yield two inequalities in the last section. 1. Introduction In the paper [5], the author consider necessary and sufficient conditions for convexity of a function x f (x) in terms of some properties of the associated function of two variables F(x, y) =(f (y) f (x))/(y x). These results are applied to the theory of the Gamma function. This paper follows the ideas in [5] and discusses the conditions for Schur-convexity and some statements of the function F(x, y) =(f (y) f (x))/(g(y) g(x)), and use them to yield some new inequalities in a triangle. For example it is shown that in an acute ABC, 1 < sin A sin B + sin B sin C B C + sin C sin A C A 3. Actually the best lower bound of the inequality above is not 1 but 4 π. The similar case is happened to some other inequalities. In the fourth section, we prove two theorems to find the optimal lower or upper bound of each inequality. Firstly, we recall the concepts of majorization and Schur-convexity. For any x =(x 1, x,, x n ) R n,let x [1] x [] x [n], denote the components of x in decreasing order. if DEFINITION. The vector x is said to be majorized by the vector y (denoted x y ) k x [i] k y [i], k = 1,,, n 1, (1.1) Mathematics subject classification (000): 6D05; 5A40. Key words and phrases: Schur-Convexity, Majorization, Trigonometric Inequality. Supported by Educational Foundation of Sichuan province, P.R.China, [005]198. c D l,zagreb Paper JMI

2 430 KUN ZHU, HONG ZHANG AND KAIQING WENG and n x [i] = n y i. (1.) DEFINITION. Ifx y but x is not a permutation of y,then x is said to be strictly majorized by y (denoted x y ). DEFINITION. A function ϕ : R n R is called Schur-convex if x y ϕ(x) ϕ(y). If, in addition, ϕ(x) < ϕ(x) whenever x y,thenϕ(x) is said to be strictly Schur-convex. Of course ϕ(x) is said to be (strictly) Schur-concave if and only if ϕ(x) is (strictly) Schur-convex.. Conditions for Schur-convexity of the function Let f, g be functions defined on an interval I and their derivative f, g exist, and g 0. Define the function F of two variables by F(x, y) = f (y) f (x) g(y) g(x), (x y), F(x, x) = f (x) g (x), (.1) where (x, y) I. Let us now consider the following statements: (A) f is convex, g is concave, f 0andg > 0onI ;or f is concave, g is convex, f 0andg < 0onI, (B) F(x, y) f ( x+y ) g ( x+y for all x, y I, ) (C) F(x, y) f (x)+f (y) g (x)+g for all x, y I, (y) (D) F is Schur-convex on I, and (A ) f is concave, g is convex, f 0andg > 0onI ;or f is convex, g is concave, f 0andg < 0onI, (B ) F(x, y) f ( x+y ) g ( x+y for all x, y I, ) (C ) F(x, y) f (x)+f (y) g (x)+g for all x, y I, (y) (D ) F is Schur-concave on I. THEOREM.1. if f (t), g (t) is continuous on I, 1. if (A) holds then the conditions (B) (D) hold;. if (A ) holds then the conditions (B ) (D ) hold. Proof. (A) (C) :Letf is convex, g is concave and x, y I, x < y. Then for each t [x, y] we have t = λ (t)x +(1 λ (t))y,whereλ (t) =(y t)/(y x).

3 A CLASS OF NEW TRIGONOMETRIC INEQUALITIES AND THEIR SHARPENINGS 431 By Jensen s inequality, we have f (t) f (x)λ (t)+f (y(1 λ (t))). The integration over t [x, y] yields y which is equivalent to x f (t)dt f (x) y x λ (t)dt + f (y) y x [1 λ (t)]dt, (.) f (y) f (x) f (x)+f (y) (y x). (.3) Similarly, g(y) g(x) g (x)+g (y) (y x). (.4) And if f 0, g > 0onI,then f (y) f (x) 0andg(y) g(x) > 0.Weget f (y) f (x) g(y) g(x) f (x)+f (y) g (x)+g (y), (.5) which is the statement (C). The second part follows upon replacing f by f and g by g. (C) (D) : It s evident that F(x, y) is symmetric on I.Since F(x, y) x F(x, y) y = g (x)(f (y) f (x)) f (x)(g(y) g(x)) (g(y) g(x)), (.6) = f (y)(g(y) g(x)) g (y)(f (y) f (x)) (g(y) g(x)), (.7) we see that the partial derivatives are continuous in each point (x, y) I, x y.let x < y,thenf(x, y) is Schur-convex on I if and only if ([4,3.A.4]) F(x,y) y F(x,y) x 0, which is equivalent to (C). (D) (B) : Suppose that F is Schur-convex, for sufficiently small ε > 0 it follows that ( x+y ε, x+y + ε) (x, y),so F( x + y Letting ε 0,weget f ( x+y ) ε, x + y + ε) F(x, y). (.8) g ( x+y F(x, y), which is the statement (B). ) This ends the proof of (1). The proof of () is similar to (1). Let us now consider the other four statements: (A 1 ) g is concave on I, (A 1 ) g is convex on I, (A ) f is convex on I, (A ) f is concave on I. THEOREM.. if g (t) is continuous on I and f (t) =t, then the conditions (A 1 ), (B), (C), (D) are equivalent, and the conditions (A 1 ), (B ), (C ), (D ) are equivalent.

4 43 KUN ZHU, HONG ZHANG AND KAIQING WENG Proof. If g is concave on I,let x, y(x < y) be arbitrary points in I.ByTaylor s expansion around c =(x + y)/,wehave g(x) =g(c)+g (c)(x c)+ g (c) (x c) + g (ξ 1 )(x c) 3, (.9) g(y) =g(c)+g (c)(y c)+ g (c) (y c) + g (ξ )(y c) 3, (.10) where ξ 1 (x, c), ξ (c, y) and g (ξ 1 )(x c) 3 0, g (ξ )(y c) 3 0. Then from (.9) and (.10) we get g(y) g(x) =g (c)(y x)+g (ξ )(y c) 3 g (ξ 1 )(x c) 3 g (c)(y x). (.11) y x g(y) g(x) 1 g (c) Therefore wherefrom (B) follows. (B) (A 1 ) : Suppose that (B) holds and that (A 1 ) does not hold. Therefore, there exists t I with g (t) < 0. By continuity of g, there is an interval I I so g (t) < 0fort I. Then the function g is convex on I and by the above proof of (A 1 ) (B), we conclude that (B) holds for g on I, hence (B) does not hold for g, which is a contradiction. (A 1 ) (C) : Similar to prove (A) (C),ifg is concave on I,then(.4) holds. (.4) is equivalent to (C). The proofs of (C) (D) and (D) (B) are the same with theorem.1. This is the end of the proof of the first part. The second part follows upon replacing g by g. Similarly we have the theorem.3, which has been proved by Milan Merkle[5]. THEOREM.3. if f (t) is continuous on I and g(t) =t, then the conditions (A ), (B), (C), (D) are equivalent, and the conditions (A ), (B ), (C ), (D ) are equivalent. 3. Some new trigonometric inequalities A, B, C are the angles of an acute triangle ABC. PROPOSITION < sin A sin B + sin B sin C B C + sin C sin A C A 3 (3.1) Proof. Let f (t) =sin t, g(t) =t and I =(0, π ),then f = cos t is convex on I. From theorem.3, we get cos A + cos B Similarly we have cos B + cos C sin A sin B sin B sin C B C cos A + B sin A, = sin C.

5 A CLASS OF NEW TRIGONOMETRIC INEQUALITIES AND THEIR SHARPENINGS 433 and cos C + cos A sin C sin A sin B C A. Adding the three inequalities upon, it yields cos A sin A sin B sin A. cos A > 1 had been shown in [1] and sin A 3 in [3]. Similarly we can get PROPOSITION cos A csc A csc A And using theorem., we get cos A cos B tan A tan B cot A cot B sin A < 0, (3.) sec A, (3.3) sec A 4. (3.4) PROPOSITION csc A sin A sin B cos A + cos B (3.5) sin A + sin B cos A cos B sec A 3, (3.6) sec A + sec B tan A tan B sin A < 1, (3.7) 9 4 cos A cot A cot B csc A + csc B. (3.8) Using theorem.1, we have PROPOSITION 3.3 cos A + cos B sec A + sec B sin A sin B tan A tan B sin 3 A, (3.9) 4 sec A + sec B cos A + cos B tan A tan B sin A sin B csc 3 A, (3.10)

6 434 KUN ZHU, HONG ZHANG AND KAIQING WENG cos 3 A sec 3 A cos A cos B cot A cot B sin A + sin B csc A + csc B, (3.11) cot A cot B cos A cos B csc A + csc B sin A + sin B. (3.1) Till this section, we have found 1 new inequalities. But some of these inequalities don t have the best lower or upper bounds. This problem will be solved in the next two sections. 4. Theorems for Symmetric Function In this section, we prove the theorem of a kind of symmetric function. Moreover, improvement of certain known results is also presented. LEMMA 4.1. (Schur, 193) If I R is an interval and g : I R is convex, then ϕ(x) = n g(x i ) is Schur-convex on I n. Define the symmetric function G of three variablesby G(x, y, z) = F(x, y) =F(x, y)+f(y, z)+f(z, x) where (x, y, z) I 3. THEOREM 4.. Let x + y + z = s > 0, x, y, z I =(0, s ), 1. if F is Schur-convex on I, φ 1 (t) =F(t, t) is convex and φ (t) =F( s, t) is strictly convex on I, then G( s 3, s 3, s 3 ) G(x, y, z) < G( s, s, 0) ;. if F is Schur-concave on I, φ 1 (t) =F(t, t) is concave and φ (t) =F( s, t) is strictly concave on I, then G( s 3, s 3, s 3 ) G(x, y, z) > G( s, s, 0). Proof. (1). It is evident that ( s z, s z ) (x, y) ( s, s Schur-convex on I,weget F( s z, s z ) F(x, y) F( s, s ) ( s z F, s z G(x, y, z) F( s, s z). z) and F(x, y) is z). Therefore Let X 1 = s x, Y 1 = s y, Z 1 = s z,then( s 3, s 3, s 3 ) (X 1, Y 1, Z 1 ) and φ 1 (Z 1 ). By the lemma 4.1, we know that get φ 1 (Z 1 ) 3φ 1 ( s 3 )= F ( s z, ) s z = φ 1 (Z 1 ) is Schur-convex on I 3.We F( s 3, s 3 )=G( s 3, s 3, s 3 ),

7 A CLASS OF NEW TRIGONOMETRIC INEQUALITIES AND THEIR SHARPENINGS 435 so G( s 3, s 3, s 3 ) G(x, y, z). Similarly, let X = s x, Y = s y, Z = s z,then (X, Y, Z ) ( s, 0, 0) and F ( s, s z) = φ (Z ),wehave φ (Z ) < φ ( s )+φ (0) =F( s, 0)+F( s, s )=G( s, s, 0), so G(x, y, z) < G( s, s, 0). This is end of the proof of (1). The() follows upon replacing F by F. Actually, from above, if F is strictly Schur-convex (concave),and φ (t) is convex (concave), the inequalities are also satisfied. By a similar proof as Theorem 4. we get THEOREM 4.3. Let x + y + z = s > 0, x, y, z I =(0, s), 1. if F is Schur-convex on I, φ 1 (t) =F(t, t) is convex and φ (t) =F(t, 0) is strictly convex on I, then G( s 3, s 3, s 3 ) G(x, y, z) < G(s, 0, 0) ;. if F is Schur-concave on I, φ 1 (t) =F(t, t) is concave and φ (t) =F(t, 0) is strictly concave on I, then G( s 3, s 3, s 3 ) G(x, y, z) > G(s, 0, 0) This theorem can be formulated for a function in n variables. THEOREM 4.4. Define the function G 1 of n variables by G 1 (x) =F 1 (x 1,, x n 1 )+F 1 (x,, x n )+ + 1 F(x n, x 1,, x n ) Let x I n =[0, s] n,s= n x i > 0, 1. if F 1 is Schur-convex on I n 1, φ 1 (t) =F 1 (t, t,, t) is convex and φ (t) = F 1 (t, 0,, 0) is convex on I, then G 1 ( s n, s n,, s n ) G 1(x) < G 1 (s, 0,, 0) ;. if F 1 is Schur-concave on I n 1, φ 1 (t) =F 1 (t, t,, t) is concave and φ (t) = F 1 (t, 0,, 0) is concave on I, then G 1 ( s n, s n,, s n ) G 1(x) G 1 (s, 0,, 0). 5. Sharpening Some Inequalities In this section we will use the theorem 4. and 4.3 to find some inequalities optimal bounds. From [5] we know that sin x sin y, x y F(x, y) = x y cos x, x = y is Schur-concave on (0, π ). It is easy to prove that F( π, t) is strictly concave on (0, π ). Hence by theorem 4. we get G( π, π, 0) < G(x, y, z) G( π 3, π 3, π 3 ),whichis equivalent to the inequality (5.1). Similarly we can prove the inequalities (5.)-(5.6). PROPOSITION 5.1. In an acute ABC 4 π < sin A sin B 3. (5.1)

8 436 KUN ZHU, HONG ZHANG AND KAIQING WENG 3 3 cos A cos B < 1 4 π, (5.) 0 < tan A tan B 3 4, (5.3) 9 4 < 1, cot A cot B (5.4) 0 < sin A sin B tan A tan B 3 8, (5.5) 1 < cos A cos B cot A cot B (5.6) And using theorem 4.3 we get PROPOSITION 5.. In an arbitrary ABC, 3 3 cos A cos B < 4 π, (5.7) cos A cos B 3. (5.8) REFERENCES [1] O. BOTTEMA, R.Z.DJORDJEVIĆ, R.R.JANIĆ, D.S.MITRINOVIĆ, P.M.VASIĆ, Geometric Inequalities, Wolters-Noordhoff Groningen, translated into Chinese by Zun San, 16 (1969). [] B. C. CARLSON, The Logarithmic Mean., Amer. Math. Monthly, 79 (197), [3] J. M. CHILD, Unknown Title., Math. Gazette, 3 (1939), [4] A. W. MARSHALL, I. OLKIN, Inequalities: Theory of Majorization and Its Applications., Academic Press, New York, 7 (1979), 64. [5] MILAN MERKLE, Conditions for Convexity of A Derivative and Applications to The Gamma and Digamma Function, AFacta Universitatis, Ser. Math. Inform., 16 (001), (Received June 1, 007) (Revised May 7, 008) Kun Zhu College of Mathematics and Software Science Sichuan Normal University Chengdu China kunn@163.com Hong Zhang College of Mathematics and Software Science Sichuan Normal University Chengdu China Kaiqing Weng College of Mathematics and Software Science Sichuan Normal University Chengdu China Journal of Mathematical Inequalities jmi@ele-math.com