Sharpness of Wilker and Huygens type inequalities
© Chen and Cheung; licensee Springer. 2012
Received: 29 June 2011
Accepted: 28 March 2012
Published: 28 March 2012
We present an elementary proof of Wilker's inequality involving trigonometric functions, and establish sharp Wilker and Huygens type inequalities.
Mathematics Subject Classification 2010: 26D05.
- (a)Prove that if 0 < x < π/2, then(1)
- (b)Find the largest constant c such that
for 0 < x < π/2.
where the constants and are best possible, was also established.
Wilker type inequalities (1) and (2) have attracted much interest of many mathematicians and have motivated a large number of research papers involving different proofs and various generalizations and improvements (cf. [2–13] and the references cited therein). A brief survey of some old and new inequalities associated with trigonometric functions can be found in . These include (among other results) Wilker's inequality.
in terms of the arithmetic, geometric and harmonic means.
In this article, we present an elementary proof of Wilker's inequality (2), and we establish sharp Wilker and Huygens type inequalities.
The following lemma is also needed in the sequel.
Lemma 1.  Let a n ∈ ℝ and b n > 0, n = 0,1, 2,... be real numbers with being strictly increasing (respectively, decreasing). If the power series and are convergent for |x| < R, then the function A(x)/B(x) is strictly increasing (respectively, decreasing) on (0, R).
2. An elementary proof of Wilker's inequality (2)
for all , with the constants and being best possible. This completes the proof of (2).
3. Sharp Wilker's inequality
Motivated by (12), we are now in a position to establish our first main result.
- (ii)For 0 < x < π/2, we have(14)
The constants and are best possible.
Proof. We only prove inequality (13). The proof of (14) is analogous.
for all , with the constants and being best possible. This completes the proof of (13).
Remark 1. Inequality (14) is sharper than (13). On the other hand, there is no strict comparison between inequalities (2) and (13). There is no strict comparison between inequalities (2) and (14) either.
In view of inequalities (13) and (14), we propose the following conjecture.
4. Sharp the Wu-Srivastava inequality
Motivated by (16), we establish our second main result:
- (ii)For 0 < x < π/2, we have(18)
The constant is best possible.
Proof. We only prove inequality (18). The proof of (17) is analogous.
The proof of the inequality (21) is not difficult, and is left with the readers. This proves the claim.
with the constant being best possible. This completes the proof of (18).
In view of inequalities (17) and (18), we propose the following conjecture.
5. Skarp Huygens inequality
Motivated by (23), we establish our third main result:
- (ii)For 0 < x < π/2, we have(25)
The constants and are best possible.
Proof. We only prove inequality (25). The proof of (24) is analogous.
for all with the constants and being possible. This completes the proof of (25).
Remark 2. There is no strict comparison between inequalities (24) and (25).
In view of inequalities (24) and (25), we propose the following conjecture.
The research is supported in part by the Research Grants Council of the Hong Kong SAR, Project No. HKU7016/07P
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