The natural algorithmic approach of mixed trigonometric polynomial problems

R E S E A R C H

Open Access

The natural algorithmic approach of

mixed trigonometric-polynomial problems

Tatjana Lutovac

_{, Branko Maleševi´c}

_{and Cristinel Mortici}

*_{Correspondence:}

[email protected]

1_{Faculty of Electrical Engineering,}

University of Belgrade, Bulevar kralja Aleksandra 73, Belgrade, 11000, Serbia

Full list of author information is available at the end of the article

Abstract

The aim of this paper is to present a new algorithm for proving mixed trigonometric-polynomial inequalities of the form

n

i=1

α

by reducing them to polynomial inequalities. Finally, we show the great applicability of this algorithm and, as an example, we use it to analyze some new rational (Padé) approximations of the function cos2_{xand to improve a class of inequalities by Yang.} The results of our analysis could be implemented by means of an automated proof assistant, so our work is a contribution to the library of automatic support tools for proving various analytic inequalities.

MSC: 41A10; 26D05; 68T15; 12L05; 41A58

Keywords: mixed trigonometric-polynomial functions; Taylor series; approximations; inequalities; algorithms; automated theorem proving

1 Introduction and motivation

In this paper, we propose a general computational method for reducing some inequali-ties involving trigonometric functions to the corresponding polynomial inequaliinequali-ties. Our work has been motivated by many papers [–] recently published in this area. As an example, we mention the work of Mortici [] who extended Wilker-Cusa-Huygens in-equalities using the method he calledthe natural approach method. This method consists in comparing and replacingsinxandcosxby their corresponding Taylor polynomials as follows:

s+

i=

(–)i_xi+

(i+ )! <sinx< s

i=

(–)i_xi+ (i+ )! ,

k+

i=

(–)i_xi

(i)! <cosx< k

i= (–)i_xi

(i)!

for every integers,k∈Nandx∈(,π/).

In this way, complicated trigonometric expressions can be reduced to polynomial or rational expressions, which can be, at least theoretically, easier studied (this can be done using some software for symbolic computation, such as Maple).

For example, Mortici in [] (Theorem ) proved the following inequality:

cosx–

sinx x

 > –x



, x∈

,π 

,

by intercalating the following Taylor polynomials:

cosx–

sinx x

 +x



>  – x ! +

x ! –

x ! –

x–x_!+x_! x

 +x



=

x_R(x₎ ,,,

whereR(t) = , – ,t+ t_–t_.

Letδ≤≤δ, withδ<δ. Recall that a function defined by the formula

f(x) =

n

i=

αixpicosqixsinrix, x∈(δ,δ), ()

is named a mixed trigonometric-polynomial function, denoted in the sequel by an MTP function [, ]. Here,αi∈R\ {},pi,qi,ri∈N,n∈N. Moreover, an inequality of the formf(x) >  is called a mixed trigonometric-polynomial inequality (MTP inequality).

MTP functions currently appear in the monographs on the theory of analytical inequal-ities [, ] and [], while concrete MTP inequalinequal-ities are employed in numerous engi-neering problems (see, e.g., [, ]). A large class of inequalities arising from different branches of science can be reduced to MTP inequalities.

It is notable that many of the above-mentioned analyses and treatments of MTP inequal-ities are all rather sophisticated and involve complex transformations and estimations. Al-most all approaches are designed for ’pen and paper analysis’ and many of them are ripe for automation, being formally defined in precise detail, and yet somewhat overwhelming for humans.

Notwithstanding, the development of formal methods and procedures for automated generation of proofs of analytical inequalities remains a challenging and important task of artificial intelligence and automated reasoning [, ].

The aim of this paper is to develop a new algorithm, based on the natural approach method, for proving MTP inequalities by reducing to polynomial inequalities.

Although transformation based on the natural approach method has been made by sev-eral researchers in their isolated studies, a unified approach has not been given yet. More-over, it is interesting to note that just trigonometric expressions involving odd powers of

cosxwere studied, as the natural approach method cannot be directly applicable for the functioncos_{xover the entire interval (,π/). Our aim is to extend and formalize the} ideas of the natural approach method for a wider class of trigonometric inequalities, in-cluding also those containing even powers ofcosx, with no further restrictions.

be derived if the functionGis defined in terms of the functionsx,sinxandsin(xsinxn_),

n= , , . . . (without involvingπ). On the other hand, several algorithms [, ] and [] have been developed to determine the sign and the real zeroes of a given polynomial, so that such problems can be considered decidable (see also [, ]).

Let us denote by

T_nφ,a(x) =

n

k=

φ(k)(a) k! (x–a)

k

the Taylor polynomial ofnth degree associated with the functionφ at a pointa. Here, Tφ_n,a(x) andTφ,a

n (x) represent the Taylor polynomial ofnth degree associated with the

functionφat a pointa, in the caseTnφ,a(x)≥φ(x), respectivelyTnφ,a(x)≤φ(x), for every

x∈(a,b). We will callTφ_n,a(x) andTφ,a

n (x) an upward and a downward approximation ofφ

on (a,b), respectively.

We present a new algorithm for approximating a given MTP functionf(x) by a polyno-mial functionP(x) such that

f(x) >P(x), ()

using the upward and downward Taylor approximations Tsin_n ,(x), Tsin_n ,(x), Tcos_n ,(x), Tcos_n ,(x).

2 The natural approach method and the associated algorithm

The following two lemmas [] related to the Taylor polynomials associated with sine and cosine functions will be of great help in our study.

Lemma  Let Tn(x) =

(n–)/

i=

(–)ixi+

(i+)! .

(i) Ifn= s+ ,withs∈N,then

Tn(x)≥Tn+(x)≥sinx for every≤x≤

(n+ )(n+ ); ()

and

Tn(x)≤Tn+(x)≤sinx for every–

(n+ )(n+ )≤x≤. ()

(ii) Ifn= s+ ,withs∈N,then

Tn(x)≤Tn+(x)≤sinx for every≤x≤

(n+ )(n+ ); ()

and

Tn(x)≥Tn+(x)≥sinx for every–

(n+ )(n+ )≤x≤. ()

Lemma  Let Tn(x) =

n/

i= (–)ixi

(i)! .

(i) Ifn= k,withk∈N,then Tn(x)≥Tn+(x)≥cosx

(ii) Ifn= k+ ,withk∈N,then

Tn(x)≤Tn+(x)≤cosx

for every–(n+ )(n+ )≤x≤(n+ )(n+ ). ()

According to Lemmas -, the upper bounds of the approximation intervals of the func-tionssinxandcosxareε=

√

(n+ )(n+ ) andε= √

(n+ )(n+ ), respectively. As

ε>π_ andε>π_, the results of these lemmas are valid, in particular, in the entire interval (,π_).

Lemma 

() Letn∈Nandx∈(,π_).Then

T_nsin,(x)≥.

() Lets∈N,p∈Nandx∈(,π_).Then

Tsin_s+,(x)p≤sinpx≤Tsin_s+,(x)p. Lemma  Let k∈N,p∈Nand x∈(,π_).Then

cospx≤Tcos_k,(x)p.

In contrast to the function sinxand its downward Taylor approximations, in the in-terval (,π

) the function cosx and the downward Taylor approximations T

cos, k+(x) = k+

i= (–)

i_xi

(i)! ,k∈N, require special attention as there is no downward Taylor approxi-mationTcos_k+,(x) such thatcos_x≥(Tcos,

k+(x))for everyx∈(,

π

).

We present the following results related to the problem with downward Taylor approx-imations of the cosine function.

Proposition 

() For everyk∈N,the downward Taylor approximationTcosk+,(x)is a strictly decreasing function on(,π

).

() For everyk∈N,there exists uniqueck∈(,π_)such thatTcosk+,(ck) = . () The sequence(ck)k∈N,withc=

√

,is strictly increasing andlimk→+∞ck=π_. () For everyk∈N,there existsdk∈(ck,π_)such thatcosdk=|Tcosk+,(dk)|. () The sequence(dk)k∈Nis strictly increasing andlimk→+∞dk=

π

.

Proof () The function Tcos_k+,(x) is strictly decreasing on (,π_) since, according to Lemma , (Tcos_k+,(x))= –Tsin_k+,(x)≤.

() The existence of ck follows from the fact that Tcosk+,() =  >  and T

cos, k+(

π

) <

cos(π_) = .

() The monotonicity of the sequence (ck)k∈Nis a result of the monotonicity ofT

cos, k+(x) and Lemma (ii).

The convergence of the sequence (T_ncos,(x))n∈Nimplies the convergence of the sequence

(ck)k∈Nto

π

() The function|Tcos_k+,(x)|is decreasing on (,ck) and increasing on (ck,π_). Based on

Lemma (ii), it follows that there existsdk∈(ck,π_) such thatcosdk=|Tcosk+,(dk)|.

() This statement is a consequence of the monotonicity of the sequence (ck)k∈N and the increasing monotonicity of the function|Tcos_k+,(x)|on (ck,π_).

Corollary  Let k∈Nand p∈N.Then

() cosp_{x> (T}cos,

k+(x))pfor everyx∈(,dk); () cospx< (T_cos_k+,(x))p_{for everyx∈(d}

k,π_).

Based on the above results, we have the following.

Corollary  Let k∈Nand p∈N.Then Tcosk+,(x)is not a downward approximation of the MTP functioncospx on(dk,π_).

In order to ensure the correctness of the algorithm [, ] we will develop next in the sequel, the following problem needs to be considered.

Problem For givenδ∈(,π_) andI⊆(,π_), findk∈Nsuch that for allk∈N,k≥k andx∈I

cosx≥Tcos_k+,(x). ()

Remark Ifcosxappears in odd powers only in the given MTP functionf(x), we takek= .

One of the methods to solve the problem of downward approximation of the function

cospx,p∈Nis the method of multiple angles developed in []. All degrees of the functions

sinxandcosxare eliminated from the given MTP functionf(x) through conversion into multiple-angle expressions. This removes all even degrees of the functioncosx, but then sine and cosine functions appear in the formsinκxorcosκx, whereκx∈(,κπ_) andκ∈ N. In this case, in order to use the results of Lemmas -, we are forced to choose large enough values ofk∈N such that

√

(k+ )(k+ ) >κπ_. Note that a higher value ofk implies a higher degree of the downward Taylor approximations and of the polynomial P(x) in () (for instance, see [] and []).

Several more ideas to solve the above problem are proposed and considered below under the names of Methods A-D. In the following, the numbersckanddkare those defined in

Proposition .

Method A Ifδ<π_, find the smallestk∈Nsuch thatdk∈(δ,π_). Thenk=k.

Note that Method A assumes solving a transcendental equation of the form cosx= Tcos_k+,(x) that requires numerical methods.

Method C Ifδ<π_, find the smallestk∈Nsuch thatTcosk+,(δ)≥. Thenk=k.

Note that Method B and Method C return the same output as for givenδand for every k∈Nthe following equivalence holds true:

ck∈

δ,π 

∧Tcos_k+,(ck) = 

⇐⇒ Tcos_k+,(δ)≥.

As Method B assumes determining the rootck of the downward Taylor approximation

Tcos_k+,(x) and Method C assumes checking the sign of the downward Taylor approximation at pointx=δ, it is notable that Method C presents a faster and simpler procedure.

Method D Eliminate all even degrees of the functioncosxusing the transformation

cospx= –sinxp=

p

i= (–)i

p

i

sinix. ()

Thenk= .

Note that Method D can be applied for any  <δ≤π/. Hence, if an MTP functionf(x) is considered in the whole interval (,π

), then Method D is applicable only (apart from the multiple-angle method). However, Method D implies an increase in the number of terms needed to be estimated. Let us represent a given MTP functionf in the following form:

f(x) = m

i=

αixpicoskixsinrix+f(x), ()

where there are no terms of the formcosj_{x,j∈N, inf}

(x). The elimination of all terms of the formcoski_{xfrom () using transformation () will increase the number of addends}

in (), in the general case withk+k+· · ·+km; consequently, it will increase the number of terms of the formsinx,∈N, in () needed to be estimated.

2.1 An algorithm based on the natural approach method

Let f be an MTP function and I ⊆(,π/). We concentrate on finding a polynomial

T Pf_{(x) such that for everyx∈I,}

f(x) >T Pf(x).

In this case, the associated MTP inequalityf(x) >  can be proved if we show that for every x∈I,

T Pf_{(x) > ,}

Commenton step II of the Procedure Estimation: in the general case, the addendai(x) =

–βixpi(sinx)qi(cosx)rican be estimated in one of the following three ways: (i) ai(x) = –βixpi(sinx)qi(cosx)ri≥βixpi(Tsinsi,+(x))

qi_(–Tcos,

ki (x)) ri_,

(ii) ai(x) = –βixpi(sinx)qi(cosx)ri≥βixpi(–T

sin, si+(x))

qi_(Tcos,

ki+(x)) ri_,

(iii) ai(x) = –βixpi(sinx)qi(cosx)ri≥–βixpi(T

sin, si+(x))

qi_(Tcos,

ki (x)) ri_.

Note that for fixedsi,ki,qiandri, the method (iii) generates polynomials of the smallest

degree.

We present the following characteristic [, ] for theNatural Approachalgorithm.

Theorem  The Natural Approach algorithm is correct.

intervalI⊆(,π/)), the algorithm halts with the correct output (i.e., the algorithm

re-turns the corresponding polynomial).

3 Some applications of the algorithm

We present an application of theNatural Approachalgorithm in the proof (Application  -Theorem ) of certain new rational (Padé) approximations of the functioncos_{x, as well} as in the improvement of a class of inequalities () by Yang (Application , Theorem ).

Application  Bercu [] used the Padé approximations to prove certain inequalities for trigonometric functions. Let us denote by (f(x))[m/n] the Padé approximant [m/n] of the functionf(x).

In this example we introduce a constraint of the functioncos_{xby the following Padé} approximations:

cosx_[/]=–x

_{+ x}_{– ,x}_{+ ,}

x_{+ x}_{+ ,}

and

cosx_[/]=x

_{– x}_{+ }

x_{+ x}_{+ } .

Theorem  The following inequalities hold true for every x∈(,π_):

cosx_[/]<cosx<cosx_[/]. () Proof We first prove the left-hand side inequality (). Using the computer software for symbolic computations, we can conclude that the functionG(x) = (cosx)[/]has exactly one zeroδ= . . . . in the interval (,π

). AsG() =  >  andG(

π

) = –. . . . < , we deduce that

G(x)≥ for everyx∈(,δ] ()

and

G(x) <  for everyx∈

δ,π 

. ()

Moreover,G(x) <cosxfor everyx∈(δ,π_). We prove now that

G(x) <cosx, x∈(,δ]. ()

We search a downward Taylor polynomialTcos_k+,(x) such that for everyx∈(,δ],

G(x) <

Tcos_k+,(x)<cosx. ()

We apply theNatural Approachalgorithm to the functionf(x) =cosx,x∈(,δ], to de-termine the downward Taylor polynomialTcos_k+,(x) such that

We can use Method C or Method D from theNatural Approachalgorithm sinceδ<π_. In this proof, we choose Method C.

The smallestkfor whichTcos_k+,(δ) >  isk= . Thereforek= . In theEstimation pro-cedure only step I can be applied to the (single) addendcos_{x. In this step,s}

≥ and k≥k=  should be selected. Let us selects=  andk= .aAs a result of this selection, the output of theNatural Approachalgorithm is the polynomial

T P(x) =Tcos,(x) 

=

 –x 

! + x ! –

x ! +

x ! –

x !

 .

We prove that

Tcos_,(x)–G(x) > , x∈(,δ]. ()

This is true since

Tcos_,(x)–G(x) =

x

,,,,(x_{+ x}_{+ ,)}Q(x),

where

Q(x) = x+ x, – x+ ,x, – ,x

+ ,,, – ,,x> .

Finally, we haveG(x) <cosxfor everyx∈(,δ]. According to (), we have

G(x) <cosx for everyx∈

,π 

.

Now we prove the right-hand side inequality (). ForG(x) = (cosx)[/], we prove the following inequalities for everyx∈(,π

):

cosx<T_cos,(x)<G(x). () Based on Proposition , it is enough to prove that for everyx∈(,π_),

Tcos_ x,(x)<G(x). ()

This is true as

G(x) –

Tcos_ x,(x)= x



,,,(x_{+ x}_{+ )}R(x),

where

R(x) =x, – x+x,, – ,x+ , – ,x> .

Sincecos_x≤(Tcos,

k (x)), for everyk∈Nand allx∈(,π_), we have

cosx<G(x) for everyx∈

,π 

Note Using Padé approximations, Bercu [, ] recently refined certain trigonometric in-equalities over various intervalsI= (,δ)⊆(,π

). All such inequalities can be proved in a similar way and using theNatural Approachalgorithm as in the proof of Theorem .

Application  Jang [] proved the following inequalities for everyx∈(,π):

cosx

≤

sinx

x ≤cos

x  ≤

 +cosx

 . ()

Previously, Klén et al. [] proved the above inequality on (,√/) only.

In this example we propose the following improvement of ().

Theorem  The following inequalities hold true for every x∈(,π)and a∈(,_):

cosx

≤

sinx x

a

≤sinx

x . ()

Proof Asa>  and  <sin_xx< , we have

sinx x

a

<sinx

x .

We prove now the following inequality:

cosx

<

sinx x

a

()

for everyx∈(,π) anda∈(,_). It suffices to show that the following mixed logarithmic-trigonometric-polynomial function []

F(x) =aln

sinx x

– ln

cosx



()

is positive for everyx∈(,π) anda∈(,_). Given that

lim

x→F(x) = , ()

based on the ideas from [], we connect the functionF(x) to the analysis of its derivative

F(x) = 

f(x

) xsinx_cos_x,

where

f(t) = t(a– )cost– asintcost– t(a– ). ()

The inequalityF(x) >  is equivalent tof(t) > . The proof of the later inequality will be done using theNatural Approachalgorithm for the functionf(t) on (,π_), witha∈(,_). As before, we search a polynomialT P(t) such that

f(t) >T P(t) > .

In step  of theNatural Approachalgorithm, we can use Method D only becauseδ=π_. Then

f(t) = t(a– ) –sint– asintcost– t(a– )

= t( –a)sint– asintcost+ ta ()

withk= . In theEstimationprocedure onlyb_{step II can be applied to the first and second} addends in (), wheresi≥ andki≥,i= , , should be selected. Let us, for example,

selects=k=s=k= . As a result of this selection, theNatural Approachalgorithm yields the polynomial

T P(t) = t( –a)

t– t

₊  t

 

– a

t– t

₊  t



 – t

₊  t



+ ta

for whichf(t) >T P(t), for everyt∈(,π

) anda∈(, 

). The inequalityf(t) >  is reduced to a decidable problem

T P(t) > , for everyt∈

,π 

anda∈

,



. ()

The sign of the polynomialT P(t) can be determined in several ways. For example, let us represent the polynomialT P(t) as

T P(t) =p(t)a+q(t), ()

where

p(t) = –t

_(t_{– t}_{+ ,t}_{– ,t}_{+ ,)}

,

and

q(t) = t

t– t

₊  t

 

.

For every fixedt∈(,π

), the functionT P(t) =p(t)a+q(t) is linear, monotonically de-creasing with respect toa∈(,_) since for everyt∈(,π

),

p(t) = – t 

,

Hence, for every fixedt∈(,π_), the value of () is greater than the value of the same expression fora=_:

p(t)

+q(t) = – t ,

t– t+ t– ,.

But

p(t) +q(t) =

t ,

t – t+  – t> ,

so inequality () is true; and consequently,F(x) >  on (,π) for everya∈(,_). But

lim_x→F(x) = , soF(x) >  on (,π) for everya∈(,_).

Remark on Theorem  Let us consider possible refinements of inequality () by a real analytical functionϕa(x) = (sin_xx)a forx∈(,δ) anda∈R. The functionϕa(x) is real

ana-lytical as it is related to the anaana-lytical function

t(x) =aln

sinx x =a ∞ k=

(–)k_k–_B k

k(k)! x

k _()

(Biare the Bernoulli numbers; see, e.g., []). The following consideration of the sign of

the analytical function in the left and right neighborhood of zero is based on Theorem . from []. Let us consider the real analytical function

f(x) =

sinx x

a

–cos x

 = –a +   x+

a – a –  

x+· · ·, ()

x∈(,π). The restriction

f_() = –a  +



> , ()

i.e., a∈ –∞,  , ()

is a necessary and sufficient condition forf(x) >  to hold on an interval (,δ(a)) (for some

δ(_a)> ). Also, the restriction

a∈  ,∞ ()

is a necessary and sufficient condition forf(x) <  to hold on an interval (,δ(a)) (for some

δ(_a)> ). The following equivalences hold true for everyx∈(,π):

a∈(,∞) ⇐⇒

sinx x

a

<sinx

x , ()

a∈(–∞, ) ⇐⇒ sinx

The refinement in Theorem  is given based on the possible values of the parameterain () and (). A similar analysis shows us that only the following refinements of inequality () are possible.

Corollary  Let a∈[_, +∞).There existsδ> such that for every x∈(,δ),it holds

sinx x

a

≤cosx

. ()

Corollary  Let a∈(–∞, ).There existsδ> such that for every x∈(,δ),it holds

 +cosx

 ≤

sinx x

a

. ()

4 Conclusions and future work

The results of our analysis could be implemented by means of an automated proof assistant [], so our work is a contribution to the library of automatic support tools [] for proving various analytic inequalities.

Our general algorithm associated with the natural approach method can be successfully applied to prove a wide category of classical MTP inequalities. For example, theNatural Approachalgorithm has recently been used to prove several open problems that involve MTP inequalities (see, e.g., [–]).

It is our contention that theNatural Approachalgorithm can be used to introduce and solve other new similar results. Chen [] used a similar method to prove the following inequalities, for everyx∈(, ):

 +  x

_arctanx<

arcsinx x



+arctanx x

and

 +  x

_{arctanx< }

arcsinx x

+arctanx

x ;

then he proposed the following inequalities as a conjecture:

arcsinx x



+arctanx x <  +

π_{+π– }

π x

_{arctanx, x∈(, )}

and



arcsinx x

+arctanx x <  +

π– 

π x

_{arctanx, x∈(, ).}

Very recently, Malešević et al. [] solved this open problem using the same procedure, i.e., the natural approach method, associated with upwards and downwards approximations of the inverse trigonometric functions.

onR. The power series of the functioncosn_{xis an alternating sign series. For example,}

forn=  andx∈R, we have

cosx=  –x+ x

_–  x

_{+· · ·=  +} ∞

k=

k–_(–)k

(k)! x k_.

Therefore, for the above power (Taylor) series, it is not hard to determine (depending onm) which partial sums (i.e., Taylor polynomials)Tcosx,

m (x) become good downward or

upward approximations of the functioncos_{xin a given intervalI. Assuming the following} representation of the functioncosnxin power (Taylor) series

cosnx=a_(n)–a(_n)x+a_(n)x–a_(n)x+· · ·,

witha(_j n)>  (j= , , , , . . .), the power (Taylor) series of functioncosn+_{xwill be an} alternating sign series as follows:

cosn+x=cosx·cosnx = a(_n)

a(_n+)

–a(_n)+a(_n)

a(_n+)

x

+

 a

(n)  +a

(n)  +a

(n) 

a(_n+)

x

–

 a

(n)  +

 a

(n)  +a

(n)  +a

(n) 

a(_n+)

x

+· · ·

witha(_j n+)>  (j= , , , , . . .).

Therefore, in general, for the functioncosn_{x, it is possible to determine, depending}

on the form of the real natural number m, the upward (downward) Taylor approxima-tionsTcos

n_x, m (x) (T

cosn_x,

m (x)) that are all above (below) the considered function in a given

intervalI. Such estimation of the functioncosn_{xand the use of corresponding Taylor}

approximations will be the object of future research.

Competing interests

Authors would like to state that they do not have any competing interest in subject of this research.

Authors’ contributions

All authors participated in every phase of research conducted for this paper.

Author details

1_{Faculty of Electrical Engineering, University of Belgrade, Bulevar kralja Aleksandra 73, Belgrade, 11000, Serbia.}2_Valahia

University of Târgovi¸ste, Bd. Unirii 18, Târgovi¸ste, 130082, Romania.3_{Academy of Romanian Scientists, Splaiul}

Independen¸tei 54, Bucharest, 050094, Romania.4_{University Politehnica of Bucharest, Splaiul Independen¸tei 313,}

Acknowledgements

The first and the second authors were supported in part by the Serbian Ministry of Education, Science and Technological Development, Projects TR 32023 and ON 174032, III 44006. The third author was supported by a Grant of the Romanian National Authority for Scientific Research, CNCS-UEFISCDI, with the Project Number PN-II-ID-PCE-2011-3-0087.

Endnotes

a _{For the selections1= 0 andk1= 1, the output of theNatural Approachalgorithm is the polynomial}

T P(x) =Tcos,0 6 (x) = 1 –

x2

2!+

x4

4!–

x6

6!

such thatT P(x)≶G1(x) holds for somex∈(0,δ]. b _{Because for every fixeda∈(1,}3

2):α1= 4(1 –a) < 0 andα2= –2a< 0.

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Received: 28 February 2017 Accepted: 2 May 2017

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