On generalization of D'Aurizio-Sándor trigonometric inequalities with a parameter
aa r X i v : . [ m a t h . C A ] M a r Journal ofM athematical& Inequalities
ON GENERALIZATION OF D’AURIZIO–S ´ANDORTRIGONOMETRIC INEQUALITIES WITH A PARAMETER L I -C HANG H UNG AND P EI -Y ING L I Submitted to J. Math. Inequal.Abstract.
In this work, we generalize the D’Aurizio-S´andor inequalities ([2, 4]) using an elemen-tary approach. In particular, our approach provides an alternative proof of the D’Aurizio-S´andorinequalities. Moreover, as an immediate consequence of the generalized D’Aurizio-S´andor in-equalities, we establish the D’Aurizio-S´andor-type inequalities for hyperbolic functions.
1. Introduction
Based on infinite product expansions and inequalities on series and the Riemann’szeta function, D’Aurizio ([2]) proved the following inequality:1 − cos x cos x x < π , (1)where x ∈ ( , π / ) . Using an elementary approach, S´andor ([4]) offered an alternativeproof of (1) by employing trigonometric inequalities and an auxiliary function. In thesame paper, S´andor also provided the converse to (1):1 − cos x cos x x > , (2)where x ∈ ( , π / ) . In addition, S´andor found the following analogous inequality (4)holds true for the case of sine functions:T HEOREM
1. (D’Aurizio-S´andor inequalities ([2, 4]))
The two double inequalities < − cos x cos x x < π (3) and π ( − √ ) < − sin x sin x x <
14 (4) hold for any x ∈ ( , π / ) . Mathematics subject classification (2010): 26D15, 26D99.
Keywords and phrases : Inequalities; trigonometric functions; monotonicity. − cos x cos xp x and p − sin x sin xp x by f c ( x ) and f s ( x ) , respec-tively: f cp ( x ) = − cos x cos xp x , (5) f sp ( x ) = p − sin x sin xp x . (6)Our aim is to generalize the D’Aurizio-S´andor inequalities for the case of f cp ( x ) and f sp ( x ) as follows:T HEOREM
2. (Generalized D’Aurizio-S´andor inequalities)
Let < x < π / . Thenthe two double inequalities π < − cos x cos xp x < p − p (7) and π (cid:18) p − csc (cid:18) π p (cid:19)(cid:19) < p − sin x sin xp x < p − p (8) hold for p = , , , · · · . In particular, the double inequality (8) remains true when p = while the double inequality (7) is reversed when p = . The remainder of this paper is organized as follows. Section 2 is devoted to theproof of Theorem 2 and an alternative proof of Theorem 1. In Section 3, we establishanalogue of Theorem 2 for hyperbolic functions. As an application of Theorem 2, weapply in Section 4 inequality (8) to the Chebyshev polynomials of the second kind andestablish a trigonometric inequality.
2. Proof of the main results
At first we will prove the following lemma. The lemma provides expressions ofthe higher-order derivative d dx ( x ddx f △ p ( x )) involving f △ p ( x ) ( △ = c , s ) , which arehelpful in proving Theorem 2. We note that the sign of d dx ( x ddx f △ p ( x )) plays a crucialrole in proving Theorem 2.L EMMA Let < x < π / and k = , , , · · · . Then when p ∈ R and p = ,we have i ) d dx (cid:18) x ddx f cp ( x ) (cid:19) = − x csc (cid:16) xp (cid:17) p (cid:18) ( p + ) sin (cid:16) x − xp (cid:17) + ( p − ) sin (cid:16) x + xp (cid:17) + (cid:0) p + p − p − (cid:1) sin (cid:16) x − xp (cid:17) + (cid:0) p − p − p + (cid:1) sin (cid:16) x + xp (cid:17)(cid:19) ;(9) ( ii ) d dx (cid:18) x ddx f sp ( x ) (cid:19) = x csc (cid:16) xp (cid:17) p (cid:18) ( p + ) sin (cid:16) x − xp (cid:17) − ( p − ) sin (cid:16) x + xp (cid:17) + (cid:0) − p − p + p + (cid:1) sin (cid:16) x − xp (cid:17) + (cid:0) p − p − p + (cid:1) sin (cid:16) x + xp (cid:17)(cid:19) . (10) In particular, ( iii ) when p = k ,d dx (cid:18) x ddx f sp ( x ) (cid:19) = − x k k − ∑ j = ( j + ) sin (cid:18) j + k x (cid:19) ; (11) ( iv ) when p = k + ,d dx (cid:18) x ddx f cp ( x ) (cid:19) = − x ( k + ) k ∑ j = j sin (cid:18) j k + x (cid:19) ( − ) j − , (12) d dx (cid:18) x ddx f sp ( x ) (cid:19) = − x ( k + ) k ∑ j = j sin (cid:18) j k + x (cid:19) . (13) ( v ) For △ = c , s and p ∈ R \ { } , lim x → ddx (cid:18) x ddx f △ p ( x ) (cid:19) = lim x → x ddx f △ p ( x ) = . (14) Proof. ( i ) , ( ii ) and ( v ) follows directly from calculations using elementary Cal-culus. In particular, trigonometric addition formulas are used in proving ( i ) and ( ii ) .To prove (11), we claim − x k k − ∑ j = ( j + ) sin (cid:18) j + k x (cid:19) = − x d dx sin x sin (cid:0) x k (cid:1) ! . (15)3ndeed, we rewrite14 k k − ∑ j = ( j + ) sin (cid:18) j + k x (cid:19) = d dx k − ∑ j = cos (cid:18) j + k x (cid:19)! . (16)On the other hand, making use of Euler’s formula e iz = cos z + i sin z leads to an alter-native expression of the left-hand side of (16): k − ∑ j = cos (cid:18) j + k x (cid:19) = k − ∑ j = ℜ n e i ( x k + xk j ) o = ℜ ( e i x k k − ∑ j = (cid:16) e i xk (cid:17) j ) (17) = ℜ (cid:26) e i x k − e ix − e i xk (cid:27) = ℜ ( e i x k e ix ( e − ix − e ix ) e ix k ( e − ix k − e ix k ) ) (18) = ℜ ( e ix sin (cid:0) x (cid:1) sin (cid:0) x k (cid:1) ) = cos (cid:16) x (cid:17) sin (cid:0) x (cid:1) sin (cid:0) x k (cid:1) = sin x (cid:0) x k (cid:1) , (19)where ℜ { z } is the real part of z and i = √−
1. Now it suffices to show d dx (cid:18) x ddx f sp ( x ) (cid:19) = − x d dx sin x sin (cid:0) x k (cid:1) ! . (20)Using (10) in ( ii ) , this can be achieved by straightforward calculations. Thus ( iii ) istrue. The proof of ( iv ) is similar, and we omit the details. We complete the proof ofLemma 1. (cid:3) We provide here an alternative proof of the two double inequalities in Theorem 1.
Proof. [Proof of Theorem 1] To this end, we show that for x ∈ ( , π / ) , f c ( x ) = − cos x cos x x is strictly increasing while f s ( x ) = − sin x sin x x is strictly decreasing. These lead tothe desired inequalities since it is easy to see thatlim x → f c ( x ) = , lim x → π / f c ( x ) = π , (21)lim x → f s ( x ) = , lim x → π / f s ( x ) = π ( − √ ) . (22)To see f c ( x ) is strictly increasing, we employ (9) in Lemma 1 to obtain d dx (cid:18) x ddx f c ( x ) (cid:19) = − x
64 sec (cid:16) x (cid:17) (cid:18) −
44 sin (cid:16) x (cid:17) + (cid:18) x (cid:19) + sin (cid:18) x (cid:19)(cid:19) = − x
16 sec (cid:16) x (cid:17) sin (cid:16) x (cid:17) ( cos x − )( cos x + ) > . (23)As lim x → ddx (cid:0) x ddx f c ( x ) (cid:1) = ddx (cid:0) x ddx f c ( x ) (cid:1) > x ddx f c ( x ) > ddx f c ( x ) > x → (cid:0) x ddx f c ( x ) (cid:1) = f c ( x ) isstrictly increasing. 4y using (11) in Lemma 1, we have d dx (cid:18) x ddx f s ( x ) (cid:19) = − x (cid:16) x (cid:17) < , (24)from which we infer that ddx (cid:0) x ddx f s ( x ) (cid:1) < x → ddx (cid:0) x ddx f s ( x ) (cid:1) = ( v ) of Lemma 1. Then ddx (cid:18) x ddx f s ( x ) (cid:19) < x → (cid:0) x ddx f s ( x ) (cid:1) = ( v ) of Lemma 1 yields x ddx f s ( x ) < ddx f s ( x ) < f s ( x ) is strictly decreasing. This completesthe proof of the theorem. (cid:3) We are now in the position to give the proof of Theorem 2.
Proof. [Proof of Theorem 2] The proof of the case when p = p ≥ ddx f △ p ( x ) < △ = c , s . Due to ( i ) of Lemma 1, we see that d dx (cid:0) x ddx f cp ( x ) (cid:1) < p ≥ ( ii ) of Lemma 1, we use (11) and (13) in Lemma 1 to concludethat d dx (cid:0) x ddx f sp ( x ) (cid:1) < △ = c , s , d dx (cid:18) x ddx f △ p ( x ) (cid:19) < . (26)Because of the first vanishing limit in ( v ) of Lemma 1, it follows that ddx (cid:18) x ddx f △ p ( x ) (cid:19) < , (27)which, together with the fact that the second limit in ( v ) of Lemma 1 vanishes, impliesthat x ddx f △ p ( x ) < ddx f △ p ( x ) < △ = c , s . It remains to find the followinglimits: lim x → f cp ( x ) = p − p , lim x → π / f cp ( x ) = π , (28)lim x → f sp ( x ) = p − p , lim x → π / f sp ( x ) = π (cid:18) p − csc (cid:18) π p (cid:19)(cid:19) . (29)We immediately have4 π = lim x → π / f cp ( x ) < − cos x cos xp x < lim x → f cp ( x ) = p − p (30)and 4 π (cid:18) p − csc (cid:18) π p (cid:19)(cid:19) = lim x → π / f sp ( x ) < p − sin x sin xp x < lim x → f sp ( x ) = p − p . (31)The proof is completed. (cid:3) . Generalized D’Aurizio-S´andor inequalities for hyperbolic functions In this section, we show an analogue of Theorem 2 for the case of hyperbolicfunctions holds true. Let h cp ( x ) = − cosh x cosh xp x , (32) h sp ( x ) = p − sinh x sinh xp x . (33)Following the same arguments for proving Lemma 1, it can be shown that Lemma 1with cos x , sin x and f △ p ( x ) ( △ = c , s ) replaced by cosh x , sinh x and h △ p ( x ) ( △ = c , s ) respectively, remains true. It follows that we can prove ddx h △ p ( x ) < △ = c , s as inthe proof of Theorem 2. It remains to calculate the following limits:lim x → f cp ( x ) = − p p , lim x → π / f cp ( x ) = π (cid:18) − cosh (cid:16) π (cid:17) sech (cid:18) π p (cid:19)(cid:19) , (34)lim x → f sp ( x ) = − p p , lim x → π / f sp ( x ) = π (cid:18) p − sinh (cid:16) π (cid:17) csch (cid:18) π p (cid:19)(cid:19) . (35)Thus, we have the following analogue of Theorem 2 for cosh x and sinh x .T HEOREM Let < x < π / . Then the two double inequalities π (cid:18) − cosh (cid:16) π (cid:17) sech (cid:18) π p (cid:19)(cid:19) < − cosh x cosh xp x < − p p (36) and π (cid:18) p − sinh (cid:16) π (cid:17) csch (cid:18) π p (cid:19)(cid:19) < p − sinh x sinh xp x < − p p (37) hold for p = , , , · · · . In particular, the double inequality (36) is reversed when p = while the double inequality (37) remains true when p = . . Application of the generalized D’Aurizio-S´andor inequalities to the Chebyshevpolynomials of the second kinds The first few Chebyshev polynomials of the second kind U n ( x ) ( n = , , , · · · ) are ([1, 3]) U ( x ) = , (38) U ( x ) = x , (39) U ( x ) = x − , (40) U ( x ) = x − x , (41) U ( x ) = x − x + , (42) U ( x ) = x − x + x , (43) U ( x ) = x − x + x − . (44)In this section, we apply Theorem 2 to U n ( x ) with x = cos θ . By means of the formula U n ( cos θ ) = sin (( n + ) θ ) sin θ , we obtain the following corollary.C OROLLARY Let y ∈ ( , π p ) . The double inequalityp (cid:0) ( − p ) y + (cid:1) < U p − ( cos y ) < p − π (cid:18) p − csc (cid:18) π p (cid:19)(cid:19) p y (45) holds for p = , , , , · · · .Proof. The double inequality (8) in Theorem 2 can be written as p − p − p x < sin x sin xp < p − π (cid:18) p − csc (cid:18) π p (cid:19)(cid:19) x , x ∈ ( , π / ) . (46)Letting x / p = y , we have p (cid:0) ( − p ) y + (cid:1) < sin ( p y ) sin y < p − π (cid:18) p − csc (cid:18) π p (cid:19)(cid:19) p y , y ∈ ( , π p ) . (47)Since sin ( py ) sin y = U p − ( cos y ) , the proof is completed. (cid:3) E XAMPLE
1. Letting p = − y <
64 cos y −
80 cos y +
24 cos y − < − (cid:0) − csc (cid:0) π (cid:1)(cid:1) π y , (48)where y ∈ ( , π ) ≈ ( , . ) and ( − csc ( π )) π ≈ . Acknowledgements . The authors wish to express sincere gratitude to Tom Molleefor his careful reading of the manuscript and valuable suggestions to improve the read-ability of the paper. Thanks are also due to Chiun-Chuan Chen and Mach Nguyet Minhfor the fruitful discussions. The authors are grateful to the anonymous referee for manyhelpful comments and valuable suggestions on this paper.7
E F E R E N C E S[1] M. A
BRAMOWITZ AND
I. A. S
TEGUN (E DS ), Handbook of Mathematical Functions with Formulas,Graphs, and Mathematical Tables , National Bureau of Standards, Applied Mathematics Series , 9thprinting, Washington.[2] J. D’A URIZIO , Refinements of the Shafer-Fink inequality of arbitrary uniform precision , Math. In-equal. Appl. , 4 (2014), 1487–1498.[3] T. J. R IVLIN , Chebyshev Polynomials , Wiley, New York.[4] J. S ´
ANDOR , On D’Aurizio’s trigonometric inequality , J. Math. Inequal. , 3 (2016), 885–888. Li-Chang Hung, Department of Mathematics, National Taiwan University, Taipei, Taiwane-mail: [email protected]
Pei-Ying Li, Department of Finance, National Taiwan University, Taipei, Taiwane-mail: email of the Second Author