Double-sided Taylor's approximations and their applications in theory of   trigonometric inequalities

Branko Malesevic; Tatjana Lutovac; Marija Rasajski; Bojan Banjac

arXiv:1906.04641·math.CA·June 12, 2019

Double-sided Taylor's approximations and their applications in theory of trigonometric inequalities

Branko Malesevic, Tatjana Lutovac, Marija Rasajski, Bojan Banjac

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TL;DR

This paper introduces double-sided Taylor's approximations to generalize and improve existing trigonometric inequalities, providing new bounds and insights in the field.

Contribution

It presents a novel application of double-sided Taylor's approximations to enhance and extend classical trigonometric inequalities.

Findings

01

New bounds for trigonometric functions derived

02

Generalizations of classical inequalities achieved

03

Improved inequalities with tighter bounds obtained

Abstract

In this paper the double-sided Talor's approximations are used to obtain generalisations and improvements of some trigonometric inequalities.

Equations68

\displaystyle\frac{3}{8}<\displaystyle\frac{1-\mbox{\small$\displaystyle\frac{\cos x}{\cos\frac{x}{2}}$}}{x^{2}}<\displaystyle\frac{4}{\pi^{2}},

\displaystyle\frac{3}{8}<\displaystyle\frac{1-\mbox{\small$\displaystyle\frac{\cos x}{\cos\frac{x}{2}}$}}{x^{2}}<\displaystyle\frac{4}{\pi^{2}},

\displaystyle\frac{4}{\pi^{2}}\left(2-\sqrt{2}\right)<\displaystyle\frac{2-\mbox{\small$\displaystyle\frac{\sin x}{\sin\frac{x}{2}}$}}{x^{2}}<\displaystyle\frac{1}{4},

\displaystyle\frac{4}{\pi^{2}}\left(2-\sqrt{2}\right)<\displaystyle\frac{2-\mbox{\small$\displaystyle\frac{\sin x}{\sin\frac{x}{2}}$}}{x^{2}}<\displaystyle\frac{1}{4},

T_{n}^{f, a +} (x) = k = 0 \sum n \frac{f ^{(k)} ( a + )}{k !} (x - a)^{k}, n \in N_{0},

T_{n}^{f, a +} (x) = k = 0 \sum n \frac{f ^{(k)} ( a + )}{k !} (x - a)^{k}, n \in N_{0},

\mbox{$\mathbbmsl{T}$}_{n}^{f;\,a+,\,b-}(x)=\left\{\begin{array}[]{ccl}T_{n-1}^{f,\,a+}(x)+\displaystyle\frac{1}{(b-a)^{n}}R_{n}^{f,\,a+}(b-)(x-a)^{n}&,&n\geq 1\\[10.76385pt] f(b-)&,&n=0,\end{array}\right.

\mbox{$\mathbbmsl{T}$}_{n}^{f;\,a+,\,b-}(x)=\left\{\begin{array}[]{ccl}T_{n-1}^{f,\,a+}(x)+\displaystyle\frac{1}{(b-a)^{n}}R_{n}^{f,\,a+}(b-)(x-a)^{n}&,&n\geq 1\\[10.76385pt] f(b-)&,&n=0,\end{array}\right.

R_{n}^{f, a +} (x) = f (x) - T_{n - 1}^{f, a +} (x), n \in N

R_{n}^{f, a +} (x) = f (x) - T_{n - 1}^{f, a +} (x), n \in N

{\mbox{$\mathbbmsl{R}$}}_{n}^{f;\,a+,\,b-}(x)=f(x)-\mbox{$\mathbbmsl{T}$}_{n-1}^{f;\,a+,\,b-}(x),~{}~{}n\!\in\!\mathbb{N}

{\mbox{$\mathbbmsl{R}$}}_{n}^{f;\,a+,\,b-}(x)=f(x)-\mbox{$\mathbbmsl{T}$}_{n-1}^{f;\,a+,\,b-}(x),~{}~{}n\!\in\!\mathbb{N}

T_{n}^{f,\,a+}(x)<f(x)<\mbox{$\mathbbmsl{T}$}_{n}^{f;\,a+,\,b-}(x).

T_{n}^{f,\,a+}(x)<f(x)<\mbox{$\mathbbmsl{T}$}_{n}^{f;\,a+,\,b-}(x).

{\mathop{\rm sgn}}{\Big{(}}\mbox{$\mathbbmsl{T}$}_{n}^{f,\,a+,\,b-}(x)\,-\,\mbox{$\mathbbmsl{T}$}_{n+1}^{f,\,a+,\,b-}(x){\Big{)}}={\mathop{\rm sgn}}{\Big{(}}{f(b-)\,-\,T_{n}^{f,\,a+}(b)}{\Big{)}},

{\mathop{\rm sgn}}{\Big{(}}\mbox{$\mathbbmsl{T}$}_{n}^{f,\,a+,\,b-}(x)\,-\,\mbox{$\mathbbmsl{T}$}_{n+1}^{f,\,a+,\,b-}(x){\Big{)}}={\mathop{\rm sgn}}{\Big{(}}{f(b-)\,-\,T_{n}^{f,\,a+}(b)}{\Big{)}},

f (x) = k = 0 \sum \infty c_{k} (x - a)^{k},

f (x) = k = 0 \sum \infty c_{k} (x - a)^{k},

\begin{array}[]{c}T_{0}^{f,\,a+}(x)\leq\ldots\leq T_{n}^{f,\,a+}(x)\leq T_{n+1}^{f,\,a+}(x)\leq\ldots\\[4.30554pt] \ldots\leq f(x)\leq\ldots\\[4.30554pt] \ldots\leq\mbox{$\mathbbmsl{T}$}_{n+1}^{f;\,a+,\,b-}(x)\leq\mbox{$\mathbbmsl{T}$}_{n}^{f;\,a+,\,b-}(x)\leq\ldots\leq\mbox{$\mathbbmsl{T}$}_{0}^{f;\,a+,\,b-}(x),\end{array}

\begin{array}[]{c}T_{0}^{f,\,a+}(x)\leq\ldots\leq T_{n}^{f,\,a+}(x)\leq T_{n+1}^{f,\,a+}(x)\leq\ldots\\[4.30554pt] \ldots\leq f(x)\leq\ldots\\[4.30554pt] \ldots\leq\mbox{$\mathbbmsl{T}$}_{n+1}^{f;\,a+,\,b-}(x)\leq\mbox{$\mathbbmsl{T}$}_{n}^{f;\,a+,\,b-}(x)\leq\ldots\leq\mbox{$\mathbbmsl{T}$}_{0}^{f;\,a+,\,b-}(x),\end{array}

f(x)=\left\{\begin{array}[]{ccc}\mbox{\small$\displaystyle\frac{3}{8}$}&,&x=0,\\[12.91663pt] \displaystyle\frac{1-\mbox{\small$\displaystyle\frac{\cos x}{\cos\frac{x}{2}}$}}{x^{2}}&,&x\!\in\!(0,\pi).\end{array}\right.

f(x)=\left\{\begin{array}[]{ccc}\mbox{\small$\displaystyle\frac{3}{8}$}&,&x=0,\\[12.91663pt] \displaystyle\frac{1-\mbox{\small$\displaystyle\frac{\cos x}{\cos\frac{x}{2}}$}}{x^{2}}&,&x\!\in\!(0,\pi).\end{array}\right.

1 - \frac{cos x}{cos \frac{x}{2}} = 1 + sec \frac{x}{2} - 2 cos \frac{x}{2},

1 - \frac{cos x}{cos \frac{x}{2}} = 1 + sec \frac{x}{2} - 2 cos \frac{x}{2},

\begin{array}[]{ccc}\cos t=\displaystyle\sum_{k=0}^{\infty}{\mbox{\small$\displaystyle\frac{(-1)^{k}}{(2k)!}$}\,t^{2k}}&&t\!\in\!R,\\[21.52771pt] \sec t=\displaystyle\sum_{k=0}^{\infty}{\mbox{\small$\displaystyle\frac{|E_{2k}|}{(2k)!}$}\,t^{2k}}&&t\!\in\!\left(-\mbox{\small$\displaystyle\frac{\pi}{2}$},\mbox{\small$\displaystyle\frac{\pi}{2}$}\right);\end{array}

\begin{array}[]{ccc}\cos t=\displaystyle\sum_{k=0}^{\infty}{\mbox{\small$\displaystyle\frac{(-1)^{k}}{(2k)!}$}\,t^{2k}}&&t\!\in\!R,\\[21.52771pt] \sec t=\displaystyle\sum_{k=0}^{\infty}{\mbox{\small$\displaystyle\frac{|E_{2k}|}{(2k)!}$}\,t^{2k}}&&t\!\in\!\left(-\mbox{\small$\displaystyle\frac{\pi}{2}$},\mbox{\small$\displaystyle\frac{\pi}{2}$}\right);\end{array}

f(x)=\displaystyle\sum_{k=1}^{\infty}{\mbox{\small$\displaystyle\frac{|E_{2k}|-2(-1)^{k}}{2^{2k}(2k)!}$}x^{2k-2}}

f(x)=\displaystyle\sum_{k=1}^{\infty}{\mbox{\small$\displaystyle\frac{|E_{2k}|-2(-1)^{k}}{2^{2k}(2k)!}$}x^{2k-2}}

f(x)=\mbox{\small$\displaystyle\frac{3}{8}$}+\mbox{\small$\displaystyle\frac{1}{128}$}\,x^{2}+\mbox{\small$\displaystyle\frac{7}{5120}$}\,x^{4}+\mbox{\small$\displaystyle\frac{461}{3440640}$}\,x^{6}+\mbox{\small$\displaystyle\frac{16841}{1238630400}$}\,x^{8}+\ldots

f(x)=\mbox{\small$\displaystyle\frac{3}{8}$}+\mbox{\small$\displaystyle\frac{1}{128}$}\,x^{2}+\mbox{\small$\displaystyle\frac{7}{5120}$}\,x^{4}+\mbox{\small$\displaystyle\frac{461}{3440640}$}\,x^{6}+\mbox{\small$\displaystyle\frac{16841}{1238630400}$}\,x^{8}+\ldots

c_{2k-2}=\mbox{\small$\displaystyle\displaystyle\frac{|E_{2k}|-2(-1)^{k}}{2^{2k}(2k)!}$}>0\quad\mbox{and}\quad c_{2k-1}=0,

c_{2k-2}=\mbox{\small$\displaystyle\displaystyle\frac{|E_{2k}|-2(-1)^{k}}{2^{2k}(2k)!}$}>0\quad\mbox{and}\quad c_{2k-1}=0,

f(x)=\left\{\begin{array}[]{ccc}\mbox{\small$\displaystyle\frac{3}{8}$}&,&x=0,\\[8.61108pt] \displaystyle\frac{1-\mbox{\small$\displaystyle\frac{\cos x}{\cos\frac{x}{2}}$}}{x^{2}}&,&x\!\in\!(0,\pi)\end{array}\right.

f(x)=\left\{\begin{array}[]{ccc}\mbox{\small$\displaystyle\frac{3}{8}$}&,&x=0,\\[8.61108pt] \displaystyle\frac{1-\mbox{\small$\displaystyle\frac{\cos x}{\cos\frac{x}{2}}$}}{x^{2}}&,&x\!\in\!(0,\pi)\end{array}\right.

\begin{array}[]{c}\mbox{\small$\displaystyle\frac{3}{8}$}=T_{0}^{f,\,0+}(x)\leq T_{2}^{f,\,0+}(x)\leq\ldots\leq T_{2n}^{f,\,0+}(x)\leq\ldots\\[8.61108pt] \ldots\leq f(x)\leq\ldots\\[4.30554pt] \leq\mbox{$\mathbbmsl{T}$}_{2m}^{f;\,0+,\,c-}(x)\leq\ldots\leq\mbox{$\mathbbmsl{T}$}_{2}^{f;\,0+,\,c-}(x)\leq\mbox{$\mathbbmsl{T}$}_{0}^{f;\,0+,\,c-}(x)=\left(1-\mbox{\small$\displaystyle\frac{\cos c}{\cos\frac{c}{2}}$}\right)\mbox{\Large$/$}{c^{2}}.\end{array}

\begin{array}[]{c}\mbox{\small$\displaystyle\frac{3}{8}$}=T_{0}^{f,\,0+}(x)\leq T_{2}^{f,\,0+}(x)\leq\ldots\leq T_{2n}^{f,\,0+}(x)\leq\ldots\\[8.61108pt] \ldots\leq f(x)\leq\ldots\\[4.30554pt] \leq\mbox{$\mathbbmsl{T}$}_{2m}^{f;\,0+,\,c-}(x)\leq\ldots\leq\mbox{$\mathbbmsl{T}$}_{2}^{f;\,0+,\,c-}(x)\leq\mbox{$\mathbbmsl{T}$}_{0}^{f;\,0+,\,c-}(x)=\left(1-\mbox{\small$\displaystyle\frac{\cos c}{\cos\frac{c}{2}}$}\right)\mbox{\Large$/$}{c^{2}}.\end{array}

\mbox{\small$\displaystyle\frac{3}{8}$}<\displaystyle\frac{1-\mbox{\small$\displaystyle\frac{\cos x}{\cos\frac{x}{2}}$}}{x^{2}}<\mbox{\small$\displaystyle\frac{4}{\pi^{2}}$}.

\mbox{\small$\displaystyle\frac{3}{8}$}<\displaystyle\frac{1-\mbox{\small$\displaystyle\frac{\cos x}{\cos\frac{x}{2}}$}}{x^{2}}<\mbox{\small$\displaystyle\frac{4}{\pi^{2}}$}.

\mbox{\small$\displaystyle\frac{3}{8}$}\leq T_{2}^{f,\,0+}\!(x)\!=\!\mbox{\small$\displaystyle\frac{3}{8}$}\!+\!\mbox{\small$\displaystyle\frac{1}{128}$}x^{2}\leq f(x)\leq\mbox{$\mathbbmsl{T}$}_{2}^{f;\,0+,\,\pi/2-}\!(x)\!=\!\mbox{\small$\displaystyle\frac{3}{8}$}\!+\!\left(\mbox{\small$\displaystyle\frac{16}{\pi^{4}}$}\!-\!\mbox{\small$\displaystyle\frac{3}{2\pi^{2}}$}\right)x^{2}\leq\mbox{\small$\displaystyle\frac{4}{\pi^{2}}$}.

\mbox{\small$\displaystyle\frac{3}{8}$}\leq T_{2}^{f,\,0+}\!(x)\!=\!\mbox{\small$\displaystyle\frac{3}{8}$}\!+\!\mbox{\small$\displaystyle\frac{1}{128}$}x^{2}\leq f(x)\leq\mbox{$\mathbbmsl{T}$}_{2}^{f;\,0+,\,\pi/2-}\!(x)\!=\!\mbox{\small$\displaystyle\frac{3}{8}$}\!+\!\left(\mbox{\small$\displaystyle\frac{16}{\pi^{4}}$}\!-\!\mbox{\small$\displaystyle\frac{3}{2\pi^{2}}$}\right)x^{2}\leq\mbox{\small$\displaystyle\frac{4}{\pi^{2}}$}.

x \in [0, π /2] ma x ∣ R_{3}^{f, 0 +} (x) ∣ = f (π /2) = 0.01100...

x \in [0, π /2] ma x ∣ R_{3}^{f, 0 +} (x) ∣ = f (π /2) = 0.01100...

\mathop{max}\limits_{x\in[0,\pi/2]}|\mbox{$\mathbbmsl{R}$}_{3}^{f;\,0+,\,\pi/2-}(x)|=f(1.14909...)=0.00315...\,.

\mathop{max}\limits_{x\in[0,\pi/2]}|\mbox{$\mathbbmsl{R}$}_{3}^{f;\,0+,\,\pi/2-}(x)|=f(1.14909...)=0.00315...\,.

g(x)=\left\{\begin{array}[]{ccc}\mbox{\small$\displaystyle\frac{1}{4}$}&,&x=0,\\[8.61108pt] \displaystyle\frac{1-\mbox{\small$\displaystyle\frac{\sin x}{\sin\frac{x}{2}}$}}{x^{2}}&,&x\!\in\!(0,\beta].\end{array}\right.

g(x)=\left\{\begin{array}[]{ccc}\mbox{\small$\displaystyle\frac{1}{4}$}&,&x=0,\\[8.61108pt] \displaystyle\frac{1-\mbox{\small$\displaystyle\frac{\sin x}{\sin\frac{x}{2}}$}}{x^{2}}&,&x\!\in\!(0,\beta].\end{array}\right.

g (x) = g_{1} (x) - g_{2} (x)

g (x) = g_{1} (x) - g_{2} (x)

g_{1}(x)=\left\{\begin{array}[]{ccc}\mbox{\small$\displaystyle\frac{1}{4}$}&,&x=0,\\[9.68747pt] \displaystyle\frac{\cosh\!\frac{x}{2}-\cos\!\frac{x}{2}}{x^{2}}&,&x\!\in\!(0,\beta]\end{array}\right.

g_{1}(x)=\left\{\begin{array}[]{ccc}\mbox{\small$\displaystyle\frac{1}{4}$}&,&x=0,\\[9.68747pt] \displaystyle\frac{\cosh\!\frac{x}{2}-\cos\!\frac{x}{2}}{x^{2}}&,&x\!\in\!(0,\beta]\end{array}\right.

g_{2}(x)=\left\{\begin{array}[]{ccc}0&,&x=0,\\[9.68747pt] \displaystyle\frac{\cosh\!\frac{x}{2}+\cos\!\frac{x}{2}-2}{x^{2}}&,&x\!\in\!(0,\beta].\end{array}\right.

g_{2}(x)=\left\{\begin{array}[]{ccc}0&,&x=0,\\[9.68747pt] \displaystyle\frac{\cosh\!\frac{x}{2}+\cos\!\frac{x}{2}-2}{x^{2}}&,&x\!\in\!(0,\beta].\end{array}\right.

g_{1}(x)=\displaystyle\sum_{k=0}^{\infty}{\mbox{\small$\displaystyle\frac{1}{2^{4k}(4k+2)!}$}x^{4k}}

g_{1}(x)=\displaystyle\sum_{k=0}^{\infty}{\mbox{\small$\displaystyle\frac{1}{2^{4k}(4k+2)!}$}x^{4k}}

g_{2}(x)=\displaystyle\sum_{k=0}^{\infty}{\mbox{\small$\displaystyle\frac{1}{2^{4k+2}(4k+4)!}$}x^{4k+2}},

g_{2}(x)=\displaystyle\sum_{k=0}^{\infty}{\mbox{\small$\displaystyle\frac{1}{2^{4k+2}(4k+4)!}$}x^{4k+2}},

\begin{array}[]{c}\mbox{\small$\displaystyle\frac{1}{4}$}=T_{0}^{g_{1},\,0+}(x)\leq\ldots\leq T_{4n}^{g_{1},\,0+}(x)\leq T_{4n+4}^{g_{1},\,0+}(x)\leq\ldots\\[6.45831pt] \ldots\leq g_{1}(x)\leq\ldots\\[5.38193pt] \ldots\leq\mbox{$\mathbbmsl{T}$}_{4m+4}^{g_{1};\,0+,\,c-}(x)\leq\mbox{$\mathbbmsl{T}$}_{4m}^{g_{1};\,0+,\,c-}(x)\leq\ldots\leq\mbox{$\mathbbmsl{T}$}_{0}^{g_{1};\,0+,\,c-}(x)=g_{1}(c).\end{array}

\begin{array}[]{c}\mbox{\small$\displaystyle\frac{1}{4}$}=T_{0}^{g_{1},\,0+}(x)\leq\ldots\leq T_{4n}^{g_{1},\,0+}(x)\leq T_{4n+4}^{g_{1},\,0+}(x)\leq\ldots\\[6.45831pt] \ldots\leq g_{1}(x)\leq\ldots\\[5.38193pt] \ldots\leq\mbox{$\mathbbmsl{T}$}_{4m+4}^{g_{1};\,0+,\,c-}(x)\leq\mbox{$\mathbbmsl{T}$}_{4m}^{g_{1};\,0+,\,c-}(x)\leq\ldots\leq\mbox{$\mathbbmsl{T}$}_{0}^{g_{1};\,0+,\,c-}(x)=g_{1}(c).\end{array}

\begin{array}[]{c}\mbox{\small$\displaystyle\frac{1}{192}$}x^{2}=T_{2}^{g_{2},\,0+}(x)\leq\ldots\leq T_{4n+2}^{g_{2},\,0+}(x)\leq T_{4n+6}^{g_{2},\,0+}(x)\leq\ldots\\[6.45831pt] \ldots\leq g_{2}(x)\leq\ldots\\[4.30554pt] \ldots\leq\mbox{$\mathbbmsl{T}$}_{4m+6}^{g_{2};\,0+,\,c-}(x)\leq\mbox{$\mathbbmsl{T}$}_{4m+2}^{g_{2};\,0+,\,c-}(x)\leq\ldots\leq\mbox{$\mathbbmsl{T}$}_{2}^{g_{2};\,0+,\,c-}(x)=\mbox{\small$\displaystyle\frac{g_{2}(c)}{c^{2}}x^{2}$}\end{array}

\begin{array}[]{c}\mbox{\small$\displaystyle\frac{1}{192}$}x^{2}=T_{2}^{g_{2},\,0+}(x)\leq\ldots\leq T_{4n+2}^{g_{2},\,0+}(x)\leq T_{4n+6}^{g_{2},\,0+}(x)\leq\ldots\\[6.45831pt] \ldots\leq g_{2}(x)\leq\ldots\\[4.30554pt] \ldots\leq\mbox{$\mathbbmsl{T}$}_{4m+6}^{g_{2};\,0+,\,c-}(x)\leq\mbox{$\mathbbmsl{T}$}_{4m+2}^{g_{2};\,0+,\,c-}(x)\leq\ldots\leq\mbox{$\mathbbmsl{T}$}_{2}^{g_{2};\,0+,\,c-}(x)=\mbox{\small$\displaystyle\frac{g_{2}(c)}{c^{2}}x^{2}$}\end{array}

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TopicsMathematical Inequalities and Applications · Mathematics and Applications · Advanced Mathematical Theories and Applications

Full text

11institutetext: Branko Malešević, Tatjana Lutovac, Marija Rašajski and Bojan Banjac 22institutetext: *1)*School of Electrical Engineering, University of Belgrade; *2)*Faculty of Technical Sciences, University of Novi Sad

Corresponding author 22email: [email protected]

Double-sided Taylor’s approximations and their applications in theory of trigonometric inequalities

Branko Malešević ${}^{\mbox{\scriptsize 1)}}$

Tatjana Lutovac ${}^{\mbox{\scriptsize 1)}}$

Marija Rašajski ${}^{\mbox{\scriptsize 1)}}$

Bojan Banjac ${}^{\mbox{\scriptsize 2)}}$

Abstract

In this paper the double-sided Taylor’s approximations are used to obtain generalisations and improvements of some trigonometric inequalities.

1 Introduction

Many mathematical and engineering problems cannot be solved without Taylor’s approximations D_S_Mitrinovic_1970 , Milovanovic_Rassias_2014 , M_J_Cloud_B_C_Drachman_L_P_Lebedev_2014 . Particularly, their application in proving various analytic inequalities is of great importance C_Mortici_2011 , B_Malesevic_M_Makragic_JMI_2016 , Milica_Makragic_JMI_2017 , T_Lutovac_B_Malesevic_C_Mortici_JIA_2017 . Recently, numerous inequalities have been generalized and improved by the use of the so-called double-sided Taylor’s approximations Milica_Makragic_JMI_2017 , H_Alzer_M_K_Kwong_2017 , B_Malesevic_T_Lutovac_M_Rasajski_C_Mortici_Adv._Difference_Equ._2018 -M_Nenenzic_L_Zhu_AADM_2018 and B_Malesevic_M_Rasajski_T_Lutovac_2019 . Many topics regarding these approximations are presented in B_Malesevic_M_Rasajski_T_Lutovac_2019 . Some of the basic concepts and results about the double-sided Taylor’s approximations presented in B_Malesevic_M_Rasajski_T_Lutovac_2019 , which will be used in this paper, are given in the next section.

In this paper, using the double-sided Taylor’s approximations, we obtain generalizations and improvements of some trigonometric inequalities proved by J. Sandor J_Sandor_2016 .

Statement 1

$(\mbox{\rm\cite[cite]{\@@bibref{Authors Phrase1YearPhrase2}{J_Sandor_2016}{\@@citephrase{(}}{\@@citephrase{)}}}},~{}\mbox{\rm Theorem}~{}1)$ **

[TABLE]

for any $x\!\in\!(0,\pi/2)$ .

Note that J. D’Aurizio J_D_Aurizio_2014 used the infinite products as well as some inequalities connected with the Riemann zeta function $\zeta$ to prove the right-hand side inequality (1).

Statement 2

$(\mbox{\rm\cite[cite]{\@@bibref{Authors Phrase1YearPhrase2}{J_Sandor_2016}{\@@citephrase{(}}{\@@citephrase{)}}}},~{}\mbox{\rm Theorem}~{}2)$ **

[TABLE]

for any $x\!\in\!(0,\pi/2)$ .

Inequalities (1) and (2) are reducible to mixed trigonometric-polynomial inequalities and can be proved by methods and algorithms that have been developed and shown in papers B_Malesevic_M_Makragic_JMI_2016 , T_Lutovac_B_Malesevic_C_Mortici_JIA_2017 and dissertation B_D_Banjac_2019 .

In this paper, we propose and prove generalizations of inequality (1) by determining the sequence of the polynomial approximations. Also, an improvement of inequality (2) is given for some intervals. The proposed generalizations and improvements are based on the double-sided Taylor’s approximations and the corresponding results presented in B_Malesevic_M_Rasajski_T_Lutovac_2019 .

2 An overview of the results related to double-sided Taylor’s approximations

Let us consider a real function $f:(a,b)\longrightarrow\mathbb{R}$ , such that there exist finite limits $f^{(k)}(a+)=\!\lim\limits_{x\rightarrow a+}{f^{(k)}(x)}$ , for $k=0,1,\ldots,n$ .

Taylor’s polynomial

[TABLE]

and the polynomial

[TABLE]

are called the first Taylor’s approximation for the function $f$ in the right neighborhood of $a$ , and the second Taylor’s approximation for the function $f$ in the right neighborhood of $a$ , respectively.

Also, the following functions:

[TABLE]

and

[TABLE]

are called the remainder of the first Taylor’s approximation in the right neighborhood of $a$ , and the remainder of the second Taylor’s approximation in the right neighborhood of $a$ , respectively.

The following Theorem, which has been proved in S_Wu_L_Debnath_2009 and whose variants are considered in S_Wu_HM_Srivastva_2008a , S_Wu_L_Debnath_2008 and S_Wu_HM_Srivastva_2008b , provides an important result regarding Taylor’s approximations.

Theorem 2.1

$(\mbox{\rm\cite[cite]{\@@bibref{Authors Phrase1YearPhrase2}{S_Wu_L_Debnath_2009}{\@@citephrase{(}}{\@@citephrase{)}}}},~{}\mbox{\rm Theorem}~{}2)$ * Suppose that $f(x)$ is a real function on $(a,b)$ , and that $n$ is a positive integer such that $f^{(k)}(a+)$ , for $k\!\in\!\{0,1,2,\ldots,n\}$ , exist.*

Supposing that $f^{(n)}(x)$ is increasing on $(a,b)$ , then for all $x\!\in\!(a,b)$ the following inequality also holds $\,:$ **

[TABLE]

Furthermore, if $f^{(n)}(x)$ is decreasing on $(a,b)$ , then the reversed inequality of (3) holds.

The above theorem is called Theorem on double-sided Taylor’s approximations in B_Malesevic_M_Rasajski_T_Lutovac_2019 , i.e. Theorem WD in B_Malesevic_T_Lutovac_M_Rasajski_C_Mortici_Adv._Difference_Equ._2018 -M_Nenenzic_L_Zhu_AADM_2018 .

The proof of the following proposition is given in B_Malesevic_M_Rasajski_T_Lutovac_2019 .

Proposition 1

$(\mbox{\rm\cite[cite]{\@@bibref{Authors Phrase1YearPhrase2}{B_Malesevic_M_Rasajski_T_Lutovac_2019}{\@@citephrase{(}}{\@@citephrase{)}}}},~{}\mbox{\rm Proposition}~{}1)$ * Consider a real function $f:(a,b)\longrightarrow\mathbb{R}$ such that there exist its first and second Taylor’s approximations, for some $n\!\in\!{N}_{0}$ . Then,*

[TABLE]

for all $x\in(a,b)$ .

From the above proposition, as shown in B_Malesevic_M_Rasajski_T_Lutovac_2019 , the following theorem directly follows:

Theorem 2.2

$(\mbox{\rm\cite[cite]{\@@bibref{Authors Phrase1YearPhrase2}{B_Malesevic_M_Rasajski_T_Lutovac_2019}{\@@citephrase{(}}{\@@citephrase{)}}}},~{}\mbox{\rm Theorem}~{}4)$ * Consider the real analytic functions $f:(a,b)\longrightarrow\mathbb{R}$ :*

[TABLE]

where $c_{k}\!\in\!\mathbb{R}$ and $c_{k}\geq 0$ for all $k\!\in\!\mathbb{N}_{0}$ . Then,

[TABLE]

for all $x\!\in\!(a,b)$ .

3 Main results

3.1 Generalization of Statement 1

Consider the function:

[TABLE]

First, we prove that $f$ is a real analytic function on $[0,\pi)$ . Based on the elementary equality $:$

[TABLE]

and well known power series expansions I_Gradshteyn_I_Ryzhik_2014 (formula 1.411):

[TABLE]

where $E_{k}$ are Euler’s numbers I_Gradshteyn_I_Ryzhik_2014 , for $t=\mbox{\small$ \displaystyle\frac{x}{2} $}\!\in\!\left[0,\mbox{\small$ \displaystyle\frac{\pi}{2} $}\right)$ , i.e. for $x\!\in\![0,\pi)$ , we have $:$

[TABLE]

i.e.

[TABLE]

where the power series converges for $x\!\in\![0,\pi)$ .

Further, based on the elementary well-known features of Euler’s numbers $E_{k}$ , we have $:$

[TABLE]

for $k=1,2,...\,$ .

Finally, from Theorem 2.2 the following result directly follows.

Theorem 3.1

For the function

[TABLE]

and any $c\in(0,\pi)$ the following inequalities hold true:

[TABLE]

for every $x\!\in\!\left(0,c\right)$ , where $m,n\!\in\!\mathbb{N}_{0}$ .

Note that inequalities from Statement 1 can be directly obtained from (4), for $c\!=\!\mbox{\small$ \displaystyle\frac{\pi}{2} $}$

[TABLE]

Also, Theorem 3.1 gives a generalization and a sequence of improvements of results from Statement 1. For example, for $c=\pi/2$ i.e. for $x\!\in\!\left(0,\mbox{\small$ \displaystyle\frac{\pi}{2} $}\right)$ we have:

[TABLE]

Using standard numerical methods it is easy to verify:

[TABLE]

and

[TABLE]

3.2 An improvement of Statement 2

Let $\beta\!\in\!(0,\pi)$ be a fixed real number. Consider the function $:$

[TABLE]

We prove that $g$ is a real analytic function on $[0,\beta]$ .

Notice that

[TABLE]

for $x\!\in\![0,\beta]$ , where

[TABLE]

and

[TABLE]

Since the functions $g_{1}$ and $g_{2}$ are real analytic functions on $[0,\beta]$ , with the following power series expansions:

[TABLE]

and

[TABLE]

the function $g$ must also be a real analytic function on $[0,\beta]$ .

Also, from Theorem 2.2 the following results directly follow.

Theorem 3.2

For all $c\in\left(0,\pi\right)$ the following inequalities hold true $:$

[TABLE]

for all $x\!\in\!\left(0,c\right)$ , where $m,n\!\in\!\mathbb{N}_{0}$ .

Theorem 3.3

For all $c\in\left(0,\pi\right)$ the following inequalities hold true $:$

[TABLE]

for all $x\!\in\!\left(0,c\right)$ , where $m,n\!\in\!\mathbb{N}_{0}$ .

Thus, from (5), Theorem 3.2 and Theorem 3.3, for $c=\mbox{\small$ \displaystyle\frac{\pi}{2} $}$ , an improvement of inequalities from Statement 2 are obtained, as shown bellow.

First, for all $x\!\in\!\left(0,\pi/2\right)$ the following inequalities hold true $:$

[TABLE]

i.e.

[TABLE]

It is easy to check

[TABLE]

for all $x\!\in\![\delta_{2},\pi/2]$ , where $\delta_{2}=\mbox{\small$ \displaystyle\frac{4\sqrt{3}e^{-\frac{\pi}{8}}}{\pi} $}\sqrt{8+8e^{\frac{\pi}{2}}-(\pi^{2}+8\sqrt{2})e^{\frac{\pi}{4}}}=0.22525...\,$ .

Also,

[TABLE]

for all $x\!\in\![0,\delta_{1}]$ , where $\delta_{1}=\mbox{\small$ \displaystyle\frac{\sqrt{2}\pi e^{\frac{\pi}{8}}\sqrt{\pi^{2}+16\sqrt{2}-32}}{8\sqrt{(\sqrt{2}-4)e^{\frac{\pi}{4}}+e^{\frac{\pi}{2}}+1}} $}=1.55456...\,$ .

4 Conclusion

In this paper, we showed a way to prove some trigonometric inequalities using the double-sided Taylor’s approximations. The presented approach enabled generalizations of inequalities (1) i.e. produced sequences of polynomial approximations of the given trigonometric function $f$ .

Note that Theorem 2.2 cannot be applied directly to inequality (2) because the function $g$ has an alternating series expansion. We overcame this obstacle by representing this function by a linear combination of two functions whose power series expansions have nonnegative coefficients.

Our approach makes a good basis for the systematic proving of trigonometric inequalities. Developing general, automated-oriented methods for proving of trigonometric inequalities is an area our continuing interest B_Banjac_M_Nenenzic_B_Malesevic_Telfor_2015 , B_Malesevic_M_Makragic_JMI_2016 -B_Banjac_M_Makragic_B_Malesevic_Results_2016 , Milica_Makragic_JMI_2017 -B_Malesevic_I_Jovovic_B_Banjac_JMI_2017 , B_Malesevic_T_Lutovac_M_Rasajski_C_Mortici_Adv._Difference_Equ._2018 -B_Malesevic_M_Rasajski_T_Lutovac_2019 and B_D_Banjac_2019 .

Acknowledgment. Research of the first and second and third author was supported in part by the Serbian Ministry of Education, Science and Technological Development, under Projects ON 174032 & III 44006, ON 174033 and TR 32023, respectively.

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