Necessary and sufficient conditions for the two parameter generalized Wilker-type inequalities

Abstract

(ProQuest: ... denotes formulae and/or non-USASCII text omitted; see image)

In the article, we provide the necessary and sufficient conditions for the parameters [alpha] and [beta] such that the generalized Wilker-type inequality ...... holds for all .......

Sun et al. Journal of Inequalities and Applications (2016) 2016:322 DOI 10.1186/s13660-016-1270-2

http://crossmark.crossref.org/dialog/?doi=10.1186/s13660-016-1270-2&domain=pdf

Web End = Necessary and sufcient conditions for the two parameter generalized Wilker-type inequalities

Hui Sun1, Zhen-Hang Yang2 and Yu-Ming Chu1*

*Correspondence: mailto:[email protected]

Web End [email protected]

1School of Mathematics and Computation Sciences, Hunan City University, Yiyang, 413000, China Full list of author information is available at the end of the article

1 Introduction

The Wilker inequality [, ] for sine and tangent functions states that the inequality

sin xx +

tan xx > (.)

holds for all x (, /). The generalizations and improvements for the Wilker inequality

(.) have been the subject of intensive research in the recent years. Wu and Srivastava [] proved that the inequality

+

sin xx p +

tan xx q > (.)

holds for all x (, /) if > , > , q > or q min{, /}, and p q/. Baricz

and Sndor [] generalized inequality (.) to the Bessel functions.In [], Zhu proved that the inequalities

sin xx p +

tan xx p >

+

xtan x p > (.)

hold for x (, /) and p . Matejka [] presented the best possible parameter p such

that the second inequality of (.) holds for x (, /).

The Author(s) 2016. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

xsin x p +

Sun et al. Journal of Inequalities and Applications (2016) 2016:322 Page 2 of 9

Zhu [] proved that the inequalities

( ) x

sin x p +

x

tan x p < < (

) x

sin x p +

x

tan x p

are valid for all x (, /) if (p, , ) {(p, , )|p , (/)p, /}

{(p, , )| p /, /, (/)p}.

In [], Yang and Chu provided the necessary and sucient condition for the parameter such that the generalized Wilker-type inequality

+

sin xx +

tan xx > (<)

holds for any xed and all x (, /).

Very recently, Chu et al. [] proved that the two parameter generalized Wilker-type inequality

+

sin xx +

+

tan xx > (.)

holds for all x (, /) if (, ) E, and inequality (.) is reversed if (, ) E, where

E = (, )| > , > (, )| < < ,

(, )

> ,

+

+ <

(, )

,

(, )

,

E = (, )| < , + > (, )| < , + >

(, )

< ,

<

< ,

,

+ +

< (, )

<

.

The main purpose of this paper is to provide the necessary and sucient conditions for the parameters and such that the generalized Wilker-type inequality (.) and its reversed inequality hold for all x (, /).

2 Lemmas

Lemma . (See [], Lemma .) Let < < < , f, f : [, ]

R be continuous

on [, ] and dierentiable on (, ), and f (x) = on (, ). Then the inequality

f(x) f() f(x) f() > (<)

f (+)

f (+)

holds for all x (, ) if there exists (, ) such that f (x)/f (x) is strictly increasing (decreasing) on (, ) and strictly decreasing (increasing) on (, ), and

f() f() f() g() ()

f (+)

f (+) = .

Sun et al. Journal of Inequalities and Applications (2016) 2016:322 Page 3 of 9

Lemma . (See [], Lemma .) Let

R, x (, /), and F(x), G(x), H(x) and g(x) be

dened by

F(x) = cos x(sin x x cos x)(x sin x cos x), (.)

G(x) = (x sin x cos x)(sin x x cos x), (.)

H(x) = x

sin x

x +

tan xx

sin x cos x, (.)

and

g(x) =

G(x) + H(x)F(x) , (.)

respectively. Then the following statements are true:() The function g(x) is strictly increasing from (, /) onto ( + /, /) if

= .() The function g(x) is strictly increasing from (, /) onto ( + /, ) if > .

() The function g(x) is strictly decreasing from (, /) onto (, + /) if

/.

Let ,

R, x (, /) and the functions I(x), J(x) and Q,(x) be dened by

I(x) = (sinxx)

( = ), I(x) = log x log(sin x), (.)

J(x) = (tanxx)

( = ), J(x) = log(tan x) log x, (.)

and

Q,(x) =

I(x)

J(x),

respectively.Then it is not dicult to verify that

I

+ = J + = ,

Q,(x) =

I(x)

J(x) =

I(x) I(+)

J(x) J(+), (.)

Q, + =

, (.)

Q,

=

( = , < ), (.)

Q,

= lim

Q,

= log

( < ). (.)

Lemma . (See [], Lemma .) Let x (, /) and Q,(x) be dened by (.). Then the following statements are true:

Sun et al. Journal of Inequalities and Applications (2016) 2016:322 Page 4 of 9

() If + + / and , then Q,(x) is strictly decreasing on (, /).() If / and / < , then Q,(x) is strictly increasing on (, /).

() If + + / and /, then Q,(x) is strictly increasing on (, /).

Lemma . Let x (, /), Q,(x) be dened by (.) and the function x D(, ; x) be dened by

D(, ; x) = Q,(x) . (.)

Then the following statements are true:() If

R is xed and < , then there exists a unique solution = () given by

log (.)

satises the equation D(, ; ) = such that D(, ; ) > for < () and D(, ;

() =

[( ) ] (

= ), () =

) < for > ().() If < is xed, then there exists a unique solution = () satises the equation

D(, ; ) = such that D(, ; ) > for < () and D(, ; ) < for > (). In particular, one has

= () = . , =

= . . (.)

() The two functions () and () are strictly decreasing.

Proof Part () follows easily from (.)-(.) and the fact that [(/) ]/ < .() It follows from (.) and (.) that

lim

D

, ;

= , lim

D

, ;

=

. (.)

Note that

d d

( )

= ( )

log

+

> (.)

for = .

From (.), (.), and (.) we clearly see that the function D(, ; ) is strictly decreasing. Therefore, there exists a unique solution = () that satises the equation

D(, ; ) = such that D(, ; ) > for < () and D(, ; ) < for > () follows from (.) and the monotonicity of the function D(, ; ). Numerical computations show that

() = . ,

= . .

() The function () is strictly decreasing follows easily from (.) and (.). The function () is strictly decreasing due to it is the inverse function of ().

Sun et al. Journal of Inequalities and Applications (2016) 2016:322 Page 5 of 9

Lemma . Let () be dened by (.). Then

= . (.)

is the unique solution of the equation () = / / such that () < / / for < and () > / / for > .

Proof Let P() = () + / + /. Then from (.) we clearly see that

P() = ( )

( ) +

,

lim

P(

) = , lim

P(

) =

, (.)

dP()

log( ) ( ) +

[( ) ] > (.)

for = , where the last of (.) due to log x x + < for all x > with x = .

Inequality (.) implies that the function P() is strictly increasing on (, ). Therefore, there exists a unique = that satises the equation () = / /

such that () < / / for < and () > / / for > follows from (.) and the monotonicity of the function P(). Numerical computations show that

= . .

Lemma . Let Q,(x), (), and be dened by (.), (.), and (.), respectively. Then the following statements are true:

() If / = . , then the inequality Q,(x) > / holds for all x (, /) if and only if / /.

() If , then the inequality Q,(x) < / holds for all x (, /) if and only if

().() If /, then the inequality Q,(x) < / holds for all x (, /) if and only if

/ /.() If , then the inequality Q,(x) > / holds for all x (, /) if and only if

().

Proof () If / and Q,(x) > / for all x (, /), then from (.)-(.) one has

lim

x

+ x Q,(x)

d =

( )

=

lim

x

+ x

+ +

x + o x =

+ +

,

which implies that / /.

If / and / /, then we clearly see

+ +

,

. (.)

Therefore, Q,(x) > / for all x (, /) follows from Lemma .() and (.) together with (.).

Sun et al. Journal of Inequalities and Applications (2016) 2016:322 Page 6 of 9

() If and Q,(x) < / for all x (, /), then from (.) and Lemma .() we clearly see that D(, ; ) and ().

Next, we prove that Q,(x) < / for all x (, /) if and (). It follows from (.) and (.) together with the fact that

J(x) =

(tanxx)

xtan x >

for x (, /) and = that the function Q,(x) is strictly decreasing. Therefore, it suces to prove that Q,(x) < / for all x (, /) if and = ().

From (.) and Lemma .() we get

= () =

log

tan xx +

. (.)

Let = , F(x), G(x), H(x), g(x), I(x) and J(x) be dened by (.)-(.), respectively. Then simple computations lead to

I (x)

J (x) =

cos x sin x(x sin x cos x) g(x) +

F(x)x, (.)

J (x) = x

sin(x) x cos x

tan xx > (.)

for x (, /).

Let = . be dened by (.). Then it follows from Lemma .(), Lemma ., and (.) together with = . > that the function x g(x) + is strictly decreasing on (, /) and

lim

x

g(x) + = . (.)

From (.) and (.) together with the monotonicity of the function x g(x) + on the interval (, /) we clearly see that there exists x (, /) such that the function x I (x)/J (x) is strictly decreasing on (, x) and strictly increasing on (x, /).

Note that

I( ) I(+)

J()( ) J()(+)

+ g(x) + = + +

> ,

lim

x

= Q,()

= D , ();

. (.)

Therefore, Q,(x) < / for all x (, /) follows from Lemma ., (.), (.), (.), and the piecewise monotonicity of the function x I (x)/J (x) on the interval (, /).

() If / and Q,(x) < / for all x (, /), then from (.)-(.) we have

lim

x

+ x Q,(x)

+

=

=

lim

x

+ x

+ +

x + o x =

+ +

,

which implies that / /.

If / and / /, then we clearly see that

+ +

,

. (.)

Sun et al. Journal of Inequalities and Applications (2016) 2016:322 Page 7 of 9

Therefore, Q,(x) < / for x (, /) follows easily from Lemma .(), (.), and (.).

() If and Q,(x) > / for all x (, /), then (.) and Lemma .() lead to the conclusion that D(, ; ) and ().

Next, we prove that Q,(x) > / for all x (, /) if and (). Since the function Q,(x) is strictly decreasing which was proved in part (), we only need to prove that Q,(x) > / for all x (, /) if and = (). It follows from

Lemma .() and (), Lemma .(), Lemma ., and < that () = and the function g(x) + is strictly increasing on (, /) such that

lim

x

+ g(x) + = + +

+ , () > ,

+ , () = ,

=

, > ,

+ > , = .

=

which implies that /.

If and /, then Q,(x) < / for all x (, /) follows from (.) and Lemma .().

() If < and Q,(x) > / for all x (, /), then (.) and Lemma .() lead to the conclusion that D(, ; ) and ().

< , (.)

lim

x

g(x) + =

(.)

From (.), (.), and (.) we clearly see that there exists x (, /) such that the function x I (x)/J (x) is strictly increasing on (, x) and strictly decreasing on (x, /).

Therefore, Q,(x) > / for all x (, /) follows from Lemma ., (.), (.), (.), and the piecewise monotonicity of the function x I (x)/J (x) on the interval (, /).

Lemma . Let Q,(x), , and () be dened by (.) and Lemma ., respectively. Then the following statements are true:

() If , then the inequality Q,(x) < / holds for all x (, /) if and only if

/.() If < , then the inequality Q,(x) > / holds for all x (, /) if and only if

().() If /, then the inequality Q,(x) > / holds for all x (, /) if and only if

/.() If /, then the inequality Q,(x) < / holds for all x (, /) if and only if

().

Proof () If and Q,(x) < / for all x (, /), then from (.)-(.) we get

lim

x

+ x Q,(x)

lim

x

+ x

+ +

x + o x =

+ +

,

Sun et al. Journal of Inequalities and Applications (2016) 2016:322 Page 8 of 9

Next, we prove that Q,(x) > / for all x (, /) if < and (). It follows from < and () together with Lemma .() that

() = , (). (.)

Therefore, Q,(x) > / for all x (, /) follows from Lemma .() and (.). () If / and Q,(x) > / for all x (, /), then from (.)-(.) we have

lim

x

+ x Q,(x)

which implies that /.

If / and /, then Q,(x) > / for all x (, /) follows from (.) and Lemma .().

() If / and Q,(x) < / for all x (, /), then (.) and Lemma .() lead to the conclusion that D(, ; ) and ().

Next, we prove that Q,(x) < / for all x (, /) if / and (). It follows from / and () together with Lemma .() that

=

sin xx +

sin xx +

=

lim

x

+ x

+ + x + o x =

+ + ,

, (). (.)

Therefore, the desired result follows from Lemma .() and (.).

3 Main results

Let ,

R with ( + ) = and Q,(x) be dened by (.), then we clearly see that the generalized Wilker-type inequality

+

tan xx > (.)

holds for all x (, /) if and only if Q,(x) < / and ( + ) > or Q,(x) > / and

( + ) < , while the generalized Wilker-type inequality

+

+

tan xx < (.)

holds for all x (, /) if and only if Q,(x) < / and ( + ) < or Q,(x) > / and

( + ) > .

From Lemmas . and . together with inequalities (.) and (.) we get Theorems . and . immediately.

Theorem . Let ,

R with ( + ) = , (), and be dened by (.) and (.), respectively. Then the following statements are true:

() If /, then inequality (.) holds for all x (, /) if and only if

(, ) {(, )| / /, ( + ) < } and inequality (.) holds for all x (, /) if and only if (, ) {(, )| / /, ( + ) > }.

+

Sun et al. Journal of Inequalities and Applications (2016) 2016:322 Page 9 of 9

() If , then inequality (.) holds for all x (, /) if and only if

(, ) {(, )| (), ( + ) > } and inequality (.) holds for all x (, /) if and only if (, ) {(, )| (), ( + ) < }.

() If /, then inequality (.) holds for all x (, /) if and only if

(, ) {(, )| / /, ( + ) > } and inequality (.) holds for all x (, /) if and only if (, ) {(, )| / /, ( + ) < }.

() If , then inequality (.) holds for all x (, /) if and only if

(, ) {(, )| (), ( + ) < } and inequality (.) holds for all x (, /) if and only if (, ) {(, )| (), ( + ) > }.

Theorem . Let ,

R with ( +) = , , , and () be dened by Lemma .. Then the following statements are true:

() If , then inequality (.) holds for all x (, /) if and only if

(, ) {(, )| /, ( + ) > } and inequality (.) holds for all x (, /) if and only if (, ) {(, )| /, ( + ) < }.

() If < , then inequality (.) holds for all x (, /) if and only if

(, ) {(, )| (), ( + ) < } and inequality (.) holds for all x (, /) if and only if (, ) {(, )| (), ( + ) > }.

() If /, then inequality (.) holds for all x (, /) if and only if

(, ) {(, )| /, ( + ) < } {(, )| (), ( + ) > } and inequality (.) holds for all x (, /) if and only if(, ) {(, )| /, ( + ) > } {(, )| (), ( + ) < }.

Competing interests

The authors declare that they have no competing interests.

Authors contributions

All authors contributed equally to the writing of this paper. All authors read and approved the nal manuscript.

Author details

1School of Mathematics and Computation Sciences, Hunan City University, Yiyang, 413000, China. 2Department of Science and Technology, State Grid Zhejiang Electric Power Research Institute, Hangzhou, 310009, China.

Acknowledgements

The research was supported by the Natural Science Foundation of China under Grants 11371125, 61374086, and 11401191.

Received: 26 September 2016 Accepted: 2 December 2016

References

1. Wilker, JB: Problem E3306. Am. Math. Mon. 96(1), 55 (1989)2. Sumner, JS, Jagers, AA, Vowe, M, Anglesio, J: Inequalities involving trigonometric functions. Am. Math. Mon. 98(3), 264-267 (1991)

3. Wu, S-H, Srivastava, H-M: A weighted and exponential generalization of Wilkers inequality and its applications. Integral Transforms Spec. Funct. 18(7-8), 529-535 (2007)

4. Baricz, , Sndor, J: Extensions of the generalized Wilker inequality to Bessel functions. J. Math. Inequal. 2(3), 397-406 (2008)

5. Zhu, L: Some new Wilker-type inequalities for circular and hyperbolic functions. Abstr. Appl. Anal. 2009, Article ID 485842 (2009)

6. Matejka, L: Note on two Wilker-type inequalities. Int. J. Open Probl. Comput. Sci. Math. 4(1), 79-85 (2011)7. Zhu, L: A source of inequalities for circular functions. Comput. Math. Appl. 58(10), 1998-2004 (2009)8. Yang, Z-H, Chu, Y-M: Sharp Wilker-type inequalities with applications. J. Inequal. Appl. 2014, Article ID 166 (2014)9. Chu, H-H, Yang, Z-H, Chu, Y-M, Zhang, W: Generalized Wilker-type inequalities with two parameters. J. Inequal. Appl. 2016, Article ID 187 (2016)

10. Yang, Z-H, Chu, Y-M, Zhang, X-H: Sharp Cusa type inequalities with two parameters and their applications. Appl. Math. Comput. 268, 1177-1198 (2015)