Geodesic
Sharp inequalities related to the Adamović-Mitrinović, Cusa, Wilker and Huygens results
Filomat, Tome 37 (2023) no. 19, p. 6319

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DOI

In this paper, we establish sharp inequalities for trigonometric functions. For example, we consider the Wilker inequality and prove that for 0 x π/2 and n ≥ 1, 2 + ( n−1∑ j=2 d j+1x2 j+ δnx2n) x3 tan x ( sin xx )2 + tan xx 2 + ( n−1∑ j=3 d j+1x2 j+Dnx2n) x3 tan x with the best possible constants δn = dn and Dn = 2π6 − 168π4 + 15120 945π4 ( 2 π )2n − n−1∑ j=2 d j+1 ( 2 π )2n−2 j , where dk = 22k+2 ( (4k + 6) |B2k+2| + (−1)k+1 ) /(2k + 3)! and Bk are the Bernoulli numbers (k ∈N0 :=N ∪ {0}). This improves and generalizes the results given by Mortici, Nenezić and Malešević.
DOI : 10.2298/FIL2319319C
Classification : 42A10, 26D05, 26D15
Keywords: Inequalities, Double-sided Taylor’s approximations, Bernoulli numbers 
Chao-Ping Chen; Branko Malešević. Sharp inequalities related to the Adamović-Mitrinović, Cusa, Wilker and Huygens results. Filomat, Tome 37 (2023) no. 19, p. 6319 . doi: 10.2298/FIL2319319C
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     author = {Chao-Ping Chen and Branko Male\v{s}evi\'c},
     title = {Sharp inequalities related to the {Adamovi\'c-Mitrinovi\'c,} {Cusa,} {Wilker} and {Huygens} results},
     journal = {Filomat},
     pages = {6319 },
     year = {2023},
     volume = {37},
     number = {19},
     doi = {10.2298/FIL2319319C},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.2298/FIL2319319C/}
}
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