research article on the inequalities for the generalized...

Research ArticleOn the Inequalities for the Generalized Trigonometric Functions

Edward Neuman

Mathematical Research Institute 144 Hawthorn Hollow Carbondale IL 62903 USA

Correspondence should be addressed to Edward Neuman edneuman76gmailcom

Received 27 January 2014 Accepted 17 March 2014 Published 7 April 2014

Academic Editor Shusen Ding

Copyright copy 2014 Edward Neuman This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper deals with Huygens-type and Wilker-type inequalities for the generalized trigonometric functions of P Lindqvist Amajor mathematical tool used in this work is a generalized version of the Schwab-Borchardt mean introduced recently by theauthor of this work

1 Introduction

Recently the generalized trigonometric and the generalizedhyperbolic functions have attracted attention of severalresearches These functions introduced by Lindqvist in [1]depend on one parameter 119901 gt 1 They become classicaltrigonometric and hyperbolic functions when 119901 = 2It is known that they are eigenfunctions of the Dirichletproblem for the one-dimensional 119901-Laplacian For moredetails concerning a recent progress in this rapidly growingarea of functions theory the interested reader is referred to[1ndash11]

The goal of this paper is to establish some inequalitiesfor families of functions mentioned earlier in this section InSection 2 we give definitions of functions under discussionsAlso some preliminary results are included there Someuseful inequalities utilized in this note are established inSection 3 The main results involving the Huygens-type andthe Wilker-type inequalities are derived in Section 4

2 Definitions and Preliminaries

For the readerrsquos convenience we recall first definition of thecelebrated Gauss hypergeometric function 119865(120572 120573 120574 119911)

119865 (120572 120573 120574 119911) =

infin

sum

119899=0

(120572 119899) (120573 119899)

(120574 119899)

119911119899

119899 |119911| lt 1 (1)

where (120572 119899) = 120572(120572 + 1) sdot sdot sdot (120572 + 119899 minus 1) (119899 = 0) is the shiftedfactorial or Appell symbol with (120572 0) = 1 if 120572 = 0 and120574 = 0 minus1 minus2 (see eg [12])

In what follows let the parameter 119901 be strictly greaterthan 1 In some cases this assumption will be relaxed to 1 lt119901 le 2 We will adopt notation and definitions used in [5] Let

120587119901= 2

120587119901

sin (120587119901) (2)

Further let

119886119901=120587119901

2

119887119901= 2minus1119901

119865(1

1199011

119901 1 +

1

1199011

2)

119888119901= 2minus1119901

119865(11

119901 1 +

1

1199011

2)

(3)

Also let 119868 = (0 1) and let 119869 = (1infin) The generalizedtrigonometric and hyperbolic functions needed in this paperare the following homeomorphisms

sin119901 (0 119886

119901) 997888rarr 119868 cos

119901 (0 119886

119901) 997888rarr 119868

tan119901 (0 119887119901) 997888rarr 119868

sinh119901 (0 119888119901) 997888rarr 119868 cosh

119901 (0infin) 997888rarr 119869

tanh119901 (0infin) 997888rarr 119869

(4)

Hindawi Publishing CorporationInternational Journal of AnalysisVolume 2014 Article ID 319837 5 pageshttpdxdoiorg1011552014319837

2 International Journal of Analysis

The inverse functions sinminus1119901

and sinhminus1119901

are represented asfollows [7]

sinminus1119901119906 = int

119906

0

(1 minus 119905119901

)minus1119901

119889119905 = 119906119865(1

1199011

119901 1 +

1

119901 119906119901

) (5)

sinhminus1119901119906 = int

119906

0

(1 + 119905119901

)minus1119901

119889119905 = 119906119865(1

1199011

119901 1 +

1

119901 minus119906119901

)

(6)

Inverse functions of the remaining four functions can beexpressed in terms of sinminus1

119901and sinhminus1

119901 We have

cosminus1119901119906 = sinminus1

119901(119901radic1 minus 119906119901) (7)

coshminus1119901119906 = sinhminus1

119901(119901radic119906119901 minus 1) (8)

tanminus1119901119906 = sinminus1

119901(

119906

119901radic1 + 119906119901) (9)

tanhminus1119901119906 = sinhminus1

119901(

119906

119901radic1 minus 119906119901) (10)

For the later use we recall now definition of a certainbivariate mean introduced recently in [13]

119878119861119901(119909 119910) =

119901radic119910119901 minus 119909119901

cosminus1119901(119909119910)

0 le 119909 lt 119910

119901radic119909119901 minus 119910119901

coshminus1119901(119909119910)

119910 lt 119909

119909 119909 = 119910

(11)

and call 119878119861119901(119909 119910) the 119901-version of the Schwab-Borchardt

mean When 119901 = 2 the latter mean becomes a classicalSchwab-Borchardt mean which has been studied extensivelyin [14ndash20] It is clear that 119878119861

119901(119909 119910) is a nonsymmetric and

homogeneous function of degree 1 of its variablesA remarkable result states that the mean 119878119861

119901admits a

representation in terms of theGauss hypergeometric function[13]

119878119861119901(119909 119910) = 119910[119865(

1

1199011

119901 1 +

1

119901 1 minus (

119909

119910)

119901

)]

minus1

(12)

(see [13])We will need the following

Theorem A If 119909 gt 119910 then

119878119861119901(119909 119910) lt 119878119861

119901(119910 119909) (13)

Let 120572 = 1119901 and let 120573 = 1 minus 1119901 Then the inequality

[119878119861119901(119910 119909)]

120572

119910120573

lt 119878119861119901(119909 119910) (14)

holds true for all positive and unequal numbers 119909 and 119910 (see[13])

Another result of interest (see [21]) reads as follows

Theorem B Let 119906 V be positive numbers Further let 120582 ge 1and let 120583 ge 1 Then the inequality

2 lt (1

119906)

120582

+ (1

V)

120583

(15)

holds true if

119906 lt 1 lt V 119900119903 V lt 1 lt 119906 (16)

and if

1 lt120582

120582 + 120583

1

119906+

120583

120582 + 120583

1

V (17)

Also we will utilize the following result [22]

Theorem C Let 119906 V gt 0 and assume that 119906 = V If 119906V gt 1then

1

119906+1

Vlt 119906 + V (18)

3 Inequalities

The goal of this section is to establish an inequality forthe 119901-version of the Schwab-Borchardt mean 119878

119901and other

inequalities as well Applications of those results to general-ized trigonometric and generalized hyperbolic functions arepresented in the next section

We begin proving an extension of inequality (14)

Theorem 1 Let 119909 119910 gt 0 (119909 = 119910) and let 119901 gt 1 Then

119909120574

119910120575

lt [119878119861119901(119910 119909)]

120572

119910120573

lt 119878119861119901(119909 119910) (19)

where

120574 =1

119901 + 1

120575 = 1 minus 120574

120572 =1

119901

120573 = 1 minus 120572

(20)

Proof We need to prove the first inequality in (19) To thisaim we will demonstrate first that

119909120574

119910120575

lt 119878119861119901(119909 119910) (21)

This can be proven using the following upper bound forGaussrsquo hypergeometric function

119865 (119886 119887 119888 119911) lt (1 minus 119911)minus119886119887119888

(22)

which holds true if 119887 gt 0 119888 gt 119886 gt 0 and |119911| lt 1 (see [23 (34)(215)]) Application to theGauss hypergeometric function onthe right side of (12) yields

119865(1

1199011

119901 1 +

1

119901 1 minus (

119909

119910)

119901

) lt (119909

119910)

minus120574

(23)

International Journal of Analysis 3

This in conjunctionwith (12) gives the desired inequality (21)For the proof of the first inequality in (19) we apply (21) to themiddle term of (19) to obtain

[119878119861119901(119910 119909)]

120572

119910120573

gt (119910120574

119909120575

)120572

119910120573

= 119909120572120575

119910120572120574+120573

= 119909120574

119910120575

(24)

where in the last step we have used (20) The proof iscomplete

Our next result reads as follows

Theorem 2 Let 119886 and 119887 be positive and unequal numbersAlso let the number 120591 be such that

119886120591

lt 119887 (25)

where 0 lt 120591 lt 1 Then the following inequalities

1 lt 120591119887

119886+ (1 minus 120591) 119887 (26)

1 lt1

2(1198871120591minus1

+119887

119886) (27)

hold true

Proof Wewill prove now inequality (26) It follows from (25)followed by application of the inequality of arithmetic andgeometric means with weights 120591 and 1 minus 120591 that

1 lt (1

119886)

120591

119887 = (119887

119886)

120591

1198871minus120591

lt 120591119887

119886+ (1 minus 120591) 119887 (28)

Inequality (27) can be established in a similarmannerWe use(26) again followed by a little algebra to obtain

1 lt 11988711205911

119886= 1198871120591minus1

119887

119886 (29)

This yields

1 lt (1198871120591minus1

)12

(119887

119886)

12

lt1

2(1198871120591minus1

+119887

119886) (30)

where in the last step we have applied the Schwarz-Bunyakovsky inequality

4 Applications to Generalized Trigonometricand Hyperbolic Functions

In this section we present several inequalities for the gen-eralized trigonometric and hyperbolic functions Recentlyseveral inequalities for these families of functions have beenobtained We refer the interested reader to the followingpapers [3 5 7ndash9 13] and to the references therein

In order to facilitate presentation we recall first someknown results for classical trigonometric functions In par-ticular the following results

3 lt 2sin119909119909

+tan119909119909 (31)

2 lt (sin119909119909)

2

+tan119909119909

(32)

(0 lt |119909| lt 1205872) have attracted attention of severalresearchers Inequalities (31) and (32) have been obtainedrespectively by Huygens [24] andWilker [25] Several proofsof these results can be found in mathematical literature(see eg [21 22 26ndash32] and the references therein) In[22] the authors called inequalities (31) and (32) the firstHuygens and the firstWilker inequalities respectively for thetrigonometric functions

The second Huygens and the second Wilker inequalitiesfor the trigonometric functions also appear in mathematicalliterature They read respectively as follows

3 lt 2119909

sin119909+

119909

tan119909 (33)

2 lt (119909

sin119909)

2

+119909

tan119909(34)

(0 lt |119909| lt 1205872) For the proofs of the last two results theinterested reader is referred to [22 29] respectively

It is worth mentioning that there are known counterpartsof inequalities (31)ndash(34) for the hyperbolic functions Theyhave the same structure as (31)ndash(34) with following modifi-cations sin rarr sinh and tan rarr tanh The domains of theirvalidity consist of all nonzero numbers For more details andadditional references see for example [22]

We are in a position to prove the following

Theorem 3 Let 119905 isin (0 119886119901) Then

(cos119901119905)1(119901+1)

lt [

[

sin119901119905

tanhminus1119901(sin119901119905)

]

]

1119901

ltsin119901119905

119905(35)

for all 119901 gt 1

Proof Let 119909 = cos119901119905 where 119905 isin (0 119886

119901) and let 119910 = 1 By

making use of (11) and the formula

sin119901119901119905 + cos119901

119901119905 = 1 (36)

(see [7]) we obtain

119878119861119901(119909 119910) = 119878119861

119901(cos119901119905 1)

=sin119901119905

cosminus1119901(cos119901119905)

=sin119901119905

119905

(37)

Also a use of (11) followed by application of (8) yields

119878119861119901(119910 119909) = 119878119861

119901(1 cos

119901119905)

=sin119901119905

coshminus1119901(

1

cos119901119905)


=sin119901119905

sinhminus1119901(tan119901119905)

=sin119901119905

tanhminus1119901(sin119901119905)

(38)

To obtain the desired result it suffices to apply inequality (19)

A particular case of (35) when 119901 = 2 appears in [33]It is worthmentioning that the counterpart of (35) for the

generalized hyperbolic functions

(cosh119901119905)1(119901+1)

lt [

[

sinh119901119905

tanminus1119901(sinh119901119905)

]

]

1119901

ltsinh119901119905

119905 119905 gt 0

(39)

(119901 gt 1) can also be established in a similar manner Anidentity

10038161003816100381610038161003816cosh11990111990510038161003816100381610038161003816

119901

minus10038161003816100381610038161003816sinh11990111990510038161003816100381610038161003816

119901

= 1 119901 gt 1 119905 isin R (40)

needed in the proof can be found in [7] We omit furtherdetails Inequalities which connect the first and the thirdmembers of (35) and (39) have been established in [7] inTheorems 36 and 38 respectively

Our next goal is to provide short proofs of the firstHuygens and the first Wilker inequalities for the generalizedtrigonometric functions

Theorem 4 Let 119901 gt 1 and let 0 lt 119905 lt 119886119901 Then

119901 + 1 lt 119901sin119901119905

119905+tan119901119905

119905 (41)

2 lt (sin119901119905

119905)

119901

+tan119901119905

119905 (42)

Proof Wewill employTheorem 2with 119886 = cos119901119905 119887 = sin

119901119905119905

and 120591 = 1(119901+1)This yields 119887119886 = tan119901119905119905 and 1minus120591 = 119901(119901+

1) Inequality (41) follows from (26) Similarly inequality (42)is an immediate consequence of (27) because 1120591minus1 = 119901

Inequality (41) has been established by different means in[7 Theorem 316]


Theorem 5 For 1 lt 119901 le 2 the following inequalities

119901 + 1 lt 119901119905

sin119901119905+

119905

tan119901119905 (43)

2 lt (119905

sin119901119905)

119901

+119905

tan119901119905

(44)

hold true for all 119905 isin (0 119886119901)

Proof Inequality (43) is established in [7 Theorem 322] Itis included here for the sake of completeness We will provenow inequality (44) To this aim we let

119906 =sin119901119905

119905 V =

tan119901119905

119905 (45)

It follows from [7 (37)] and the proof of Lemma 332 in [7]that 119906 lt 1 lt V holds for all 119901 gt 1 To obtain the desired resultwe apply nowTheorem B with 119906 and V as defined above and120582 = 119901 and 120583 = 1 It is easy to see that inequality (17) is thesame as the second Huygens inequality (43) The assertionnow follows

The counterparts ofTheorems 4 and 5 for the generalizedhyperbolic functions can be established in a similar fashionWe leave it to the interested reader

The last result of this section gives an inequality whichconnects the first and the second inequalities of Wilker forthe generalized trigonometric functions

Theorem 6 Let 119901 gt 1 If 119905 isin (0 119886119901) then

(119905

sin119901119905)

119901

+119905

tan119901119905lt (

sin119901119905

119905)

119901

+tan119901119905

119905 (46)

Proof For the sake of notation let

119906 =sin119901119905

119905 V =

tan119901119905

119905 (47)

Then the inequality connecting the first and the third mem-bers of (35) can be written as

1 lt 119906(1

cos119901119905)

1(119901+1)

= 119906119901(119901+1)V1(119901+1) (48)

Exponentiation with the exponent of 119901 + 1 allows us to writethe inequality connecting the first and the last members as1 lt 119906

119901V To complete the proof it suffices to utilize TheoremC with 119906 replaced by 119906119901

Conflict of Interests

The author declares that he had no conflict of interests

References

[1] P Lindqvist ldquoSome remarkable sine and cosine functionsrdquoRicerche di Matematica vol 44 no 2 pp 269ndash290 1995

[2] A Baricz B A Bhayo and R Klen ldquoConvexity properties ofgeneralized trigonometric and hyperbolic functionsrdquo Aequa-tiones Mathematicae 2013

[3] B A Bhayo ldquoPowermean inequality of generalized trigonomet-ric functionsrdquo httparxivorgabs12090873

[4] B A Bhayo and M Vuorinen ldquoOn generalized trigonometricfunctions with two parametersrdquo Journal of ApproximationTheory vol 164 no 10 pp 1415ndash1426 2012

[5] B A Bhayo and M Vuorinen ldquoInequalities for eigenfunctionsof the p-Laplacianrdquo Issues of Analysis vol 2 no 20 pp 13ndash352013


[6] W-D Jiang M-K Wang Y-M Chu Y-P Jiang and F QildquoConvexity of the generalized sine function and the generalizedhyperbolic sine functionrdquo Journal of ApproximationTheory vol174 pp 1ndash9 2013

[7] R Klen M Vuorinen and X Zhang ldquoInequalities for thegeneralized trigonometric and hyperbolic functionsrdquo Journal ofMathematical Analysis and Applications vol 409 no 1 pp 521ndash529 2014

[8] E Neuman ldquoInequalities involving generalized trigonometricand generalized hyperbolic functionsrdquo Journal of MathematicalInequalities In press

[9] E Neuman ldquoWilker- and Huygens-type inequalities for thegeneralized and for the generalized hyperbolic functionsrdquoApplied Mathematics and Computation vol 230 pp 211ndash2172014

[10] S Takeuchi ldquoGeneralized Jacobian elliptic functions andtheir application to bifurcation problems associated with 119901-Laplacianrdquo Journal of Mathematical Analysis and Applicationsvol 385 no 1 pp 24ndash35 2012

[11] S Takeuchi ldquoThe basis property of generalized Jacobian ellipticfunctionsrdquo httparxivorgabs13100597

[12] B C Carlson Special Functions of Applied Mathematics Aca-demic Press New York NY USA 1977

[13] E Neuman ldquoOn the p-version of the Schwab-Borchardt meanrdquoIn press

[14] J M Borwein and P B Borwein Pi and the AGM A Study inAnalytic Number Theory and Computational Complexity JohnWiley amp Sons New York NY USA 1987

[15] J L Brenner and B C Carlson ldquoHomogeneous mean valuesweights and asymptoticsrdquo Journal of Mathematical Analysis andApplications vol 123 no 1 pp 265ndash280 1987

[16] B C Carlson ldquoA hypergeometric mean valuerdquo Proceedings ofthe American Mathematical Society vol 16 pp 759ndash766 1965

[17] B C Carlson ldquoAlgorithms involving arithmetic and geometricmeansrdquoThe American Mathematical Monthly vol 78 pp 496ndash505 1971

[18] E Neuman ldquoInequalities for the Schwab-Borchardt mean andtheir applicationsrdquo Journal of Mathematical Inequalities vol 5no 4 pp 601ndash609 2011

[19] E Neuman and J Sandor ldquoOn the Schwab-Borchardt meanrdquoMathematica Pannonica vol 14 no 2 pp 253ndash266 2003

[20] E Neuman and J Sandor ldquoOn the Schwab-Borchardt mean IIrdquoMathematica Pannonica vol 17 no 1 pp 49ndash59 2006

[21] E Neuman ldquoInequalities for weighted sums of powers and theirapplicationsrdquo Mathematical Inequalities amp Applications vol 15no 4 pp 995ndash1005 2012

[22] E Neuman and J Sandor ldquoOn some inequalities involvingtrigonometric and hyperbolic functions with emphasis on theCusa-Huygens Wilker and Huygens inequalitiesrdquo Mathemati-cal Inequalities amp Applications vol 13 no 4 pp 715ndash723 2010

[23] B C Carlson ldquoSome inequalities for hypergeometric func-tionsrdquo Proceedings of the American Mathematical Society vol17 pp 32ndash39 1966

[24] C HuygensOeuvres Completes 1888ndash1940 Societe Hollondaisedes Science Haga Gothenburg

[25] J B Wilker ldquoProblem E 3306rdquo The American MathematicalMonthly vol 96 article 55 1989

[26] B-N Guo B-M Qiao F Qi and W Li ldquoOn new proofsof Wilkerrsquos inequalities involving trigonometric functionsrdquoMathematical Inequalities amp Applications vol 6 no 1 pp 19ndash22 2003

[27] J S Sumner AA JagersMVowe and J Anglesio ldquoInequalitiesinvolving trigono-metric functionsrdquoThe American Mathemati-cal Monthly vol 98 pp 264ndash267 1991

[28] S Wu and A Baricz ldquoGeneralizations of Mitrinovic Adamovicand Lazarevicrsquos inequalities and their applicationsrdquo Publica-tiones Mathematicae Debrecen vol 75 no 3-4 pp 447ndash4582009

[29] S-H Wu and H M Srivastava ldquoA weighted and exponentialgeneralization of Wilkerrsquos inequality and its applicationsrdquo Inte-gral Transforms and Special Functions vol 18 no 7-8 pp 529ndash535 2007

[30] L Zhu ldquoA new simple proof of Wilkerrsquos inequalityrdquoMathemat-ical Inequalities amp Applications vol 8 no 4 pp 749ndash750 2005

[31] L Zhu ldquoOn Wilker-type inequalitiesrdquo Mathematical Inequali-ties amp Applications vol 10 no 4 pp 727ndash731 2007

[32] L Zhu ldquoSome new Wilker-type inequalities for circular andhyperbolic functionsrdquo Abstract and Applied Analysis vol 2009Article ID 485842 9 pages 2009

[33] E Neuman ldquoRefinements and generalizations of certaininequalities involving trigonometric and hyperbolic functionsrdquoJournal of Inequalities and Applications vol 1 no 1 pp 1ndash112012

Submit your manuscripts athttpwwwhindawicom

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Algebra

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


The inverse functions sinminus1119901

and sinhminus1119901

are represented asfollows [7]

sinminus1119901119906 = int

119906

0

(1 minus 119905119901

)minus1119901

119889119905 = 119906119865(1

1199011

119901 1 +

1

119901 119906119901

) (5)

sinhminus1119901119906 = int

119906

0

(1 + 119905119901

)minus1119901

119889119905 = 119906119865(1

1199011

119901 1 +

1

119901 minus119906119901

)

(6)

Inverse functions of the remaining four functions can beexpressed in terms of sinminus1

119901and sinhminus1

119901 We have

cosminus1119901119906 = sinminus1

119901(119901radic1 minus 119906119901) (7)

coshminus1119901119906 = sinhminus1

119901(119901radic119906119901 minus 1) (8)

tanminus1119901119906 = sinminus1

119901(

119906

119901radic1 + 119906119901) (9)

tanhminus1119901119906 = sinhminus1

119901(

119906

119901radic1 minus 119906119901) (10)

For the later use we recall now definition of a certainbivariate mean introduced recently in [13]

119878119861119901(119909 119910) =

119901radic119910119901 minus 119909119901

cosminus1119901(119909119910)

0 le 119909 lt 119910

119901radic119909119901 minus 119910119901

coshminus1119901(119909119910)

119910 lt 119909

119909 119909 = 119910

(11)

and call 119878119861119901(119909 119910) the 119901-version of the Schwab-Borchardt

mean When 119901 = 2 the latter mean becomes a classicalSchwab-Borchardt mean which has been studied extensivelyin [14ndash20] It is clear that 119878119861

119901(119909 119910) is a nonsymmetric and

homogeneous function of degree 1 of its variablesA remarkable result states that the mean 119878119861

119901admits a

representation in terms of theGauss hypergeometric function[13]

119878119861119901(119909 119910) = 119910[119865(

1

1199011

119901 1 +

1

119901 1 minus (

119909

119910)

119901

)]

minus1

(12)

(see [13])We will need the following

Theorem A If 119909 gt 119910 then

119878119861119901(119909 119910) lt 119878119861

119901(119910 119909) (13)

Let 120572 = 1119901 and let 120573 = 1 minus 1119901 Then the inequality

[119878119861119901(119910 119909)]

120572

119910120573

lt 119878119861119901(119909 119910) (14)

holds true for all positive and unequal numbers 119909 and 119910 (see[13])

Another result of interest (see [21]) reads as follows

Theorem B Let 119906 V be positive numbers Further let 120582 ge 1and let 120583 ge 1 Then the inequality

2 lt (1

119906)

120582

+ (1

V)

120583

(15)

holds true if

119906 lt 1 lt V 119900119903 V lt 1 lt 119906 (16)

and if

1 lt120582

120582 + 120583

1

119906+

120583

120582 + 120583

1

V (17)

Also we will utilize the following result [22]

Theorem C Let 119906 V gt 0 and assume that 119906 = V If 119906V gt 1then

1

119906+1

Vlt 119906 + V (18)

3 Inequalities

The goal of this section is to establish an inequality forthe 119901-version of the Schwab-Borchardt mean 119878

119901and other

inequalities as well Applications of those results to general-ized trigonometric and generalized hyperbolic functions arepresented in the next section

We begin proving an extension of inequality (14)

Theorem 1 Let 119909 119910 gt 0 (119909 = 119910) and let 119901 gt 1 Then

119909120574

119910120575

lt [119878119861119901(119910 119909)]

120572

119910120573

lt 119878119861119901(119909 119910) (19)

where

120574 =1

119901 + 1

120575 = 1 minus 120574

120572 =1

119901

120573 = 1 minus 120572

(20)

Proof We need to prove the first inequality in (19) To thisaim we will demonstrate first that

119909120574

119910120575

lt 119878119861119901(119909 119910) (21)

This can be proven using the following upper bound forGaussrsquo hypergeometric function

119865 (119886 119887 119888 119911) lt (1 minus 119911)minus119886119887119888

(22)

which holds true if 119887 gt 0 119888 gt 119886 gt 0 and |119911| lt 1 (see [23 (34)(215)]) Application to theGauss hypergeometric function onthe right side of (12) yields

119865(1

1199011

119901 1 +

1

119901 1 minus (

119909

119910)

119901

) lt (119909

119910)

minus120574

(23)



[119878119861119901(119910 119909)]

120572

119910120573

gt (119910120574

119909120575

)120572

119910120573

= 119909120572120575

119910120572120574+120573

= 119909120574

119910120575

(24)




119886120591

lt 119887 (25)


1 lt 120591119887

119886+ (1 minus 120591) 119887 (26)

1 lt1

2(1198871120591minus1

+119887

119886) (27)

hold true


1 lt (1

119886)

120591

119887 = (119887

119886)

120591

1198871minus120591

lt 120591119887

119886+ (1 minus 120591) 119887 (28)


1 lt 11988711205911

119886= 1198871120591minus1

119887

119886 (29)

This yields

1 lt (1198871120591minus1

)12

(119887

119886)

12

lt1

2(1198871120591minus1

+119887

119886) (30)





3 lt 2sin119909119909

+tan119909119909 (31)

2 lt (sin119909119909)

2

+tan119909119909

(32)



3 lt 2119909

sin119909+

119909

tan119909 (33)

2 lt (119909

sin119909)

2

+119909

tan119909(34)





(cos119901119905)1(119901+1)

lt [

[

sin119901119905

tanhminus1119901(sin119901119905)

]

]

1119901

ltsin119901119905

119905(35)

for all 119901 gt 1


119901) and let 119910 = 1 By


sin119901119901119905 + cos119901

119901119905 = 1 (36)

(see [7]) we obtain

119878119861119901(119909 119910) = 119878119861

119901(cos119901119905 1)

=sin119901119905

cosminus1119901(cos119901119905)

=sin119901119905

119905

(37)


119878119861119901(119910 119909) = 119878119861

119901(1 cos

119901119905)

=sin119901119905

coshminus1119901(

1

cos119901119905)


=sin119901119905

sinhminus1119901(tan119901119905)

=sin119901119905

tanhminus1119901(sin119901119905)

(38)




(cosh119901119905)1(119901+1)

lt [

[

sinh119901119905

tanminus1119901(sinh119901119905)

]

]

1119901

ltsinh119901119905

119905 119905 gt 0

(39)


10038161003816100381610038161003816cosh11990111990510038161003816100381610038161003816

119901

minus10038161003816100381610038161003816sinh11990111990510038161003816100381610038161003816

119901

= 1 119901 gt 1 119905 isin R (40)




119901 + 1 lt 119901sin119901119905

119905+tan119901119905

119905 (41)

2 lt (sin119901119905

119905)

119901

+tan119901119905

119905 (42)


119901119905119905






119901 + 1 lt 119901119905

sin119901119905+

119905

tan119901119905 (43)

2 lt (119905

sin119901119905)

119901

+119905

tan119901119905

(44)



119906 =sin119901119905

119905 V =

tan119901119905

119905 (45)





(119905

sin119901119905)

119901

+119905

tan119901119905lt (

sin119901119905

119905)

119901

+tan119901119905

119905 (46)


119906 =sin119901119905

119905 V =

tan119901119905

119905 (47)


1 lt 119906(1

cos119901119905)

1(119901+1)

= 119906119901(119901+1)V1(119901+1) (48)





References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











[119878119861119901(119910 119909)]

120572

119910120573

gt (119910120574

119909120575

)120572

119910120573

= 119909120572120575

119910120572120574+120573

= 119909120574

119910120575

(24)




119886120591

lt 119887 (25)


1 lt 120591119887

119886+ (1 minus 120591) 119887 (26)

1 lt1

2(1198871120591minus1

+119887

119886) (27)

hold true


1 lt (1

119886)

120591

119887 = (119887

119886)

120591

1198871minus120591

lt 120591119887

119886+ (1 minus 120591) 119887 (28)


1 lt 11988711205911

119886= 1198871120591minus1

119887

119886 (29)

This yields

1 lt (1198871120591minus1

)12

(119887

119886)

12

lt1

2(1198871120591minus1

+119887

119886) (30)





3 lt 2sin119909119909

+tan119909119909 (31)

2 lt (sin119909119909)

2

+tan119909119909

(32)



3 lt 2119909

sin119909+

119909

tan119909 (33)

2 lt (119909

sin119909)

2

+119909

tan119909(34)





(cos119901119905)1(119901+1)

lt [

[

sin119901119905

tanhminus1119901(sin119901119905)

]

]

1119901

ltsin119901119905

119905(35)

for all 119901 gt 1


119901) and let 119910 = 1 By


sin119901119901119905 + cos119901

119901119905 = 1 (36)

(see [7]) we obtain

119878119861119901(119909 119910) = 119878119861

119901(cos119901119905 1)

=sin119901119905

cosminus1119901(cos119901119905)

=sin119901119905

119905

(37)


119878119861119901(119910 119909) = 119878119861

119901(1 cos

119901119905)

=sin119901119905

coshminus1119901(

1

cos119901119905)


=sin119901119905

sinhminus1119901(tan119901119905)

=sin119901119905

tanhminus1119901(sin119901119905)

(38)




(cosh119901119905)1(119901+1)

lt [

[

sinh119901119905

tanminus1119901(sinh119901119905)

]

]

1119901

ltsinh119901119905

119905 119905 gt 0

(39)


10038161003816100381610038161003816cosh11990111990510038161003816100381610038161003816

119901

minus10038161003816100381610038161003816sinh11990111990510038161003816100381610038161003816

119901

= 1 119901 gt 1 119905 isin R (40)




119901 + 1 lt 119901sin119901119905

119905+tan119901119905

119905 (41)

2 lt (sin119901119905

119905)

119901

+tan119901119905

119905 (42)


119901119905119905






119901 + 1 lt 119901119905

sin119901119905+

119905

tan119901119905 (43)

2 lt (119905

sin119901119905)

119901

+119905

tan119901119905

(44)



119906 =sin119901119905

119905 V =

tan119901119905

119905 (45)





(119905

sin119901119905)

119901

+119905

tan119901119905lt (

sin119901119905

119905)

119901

+tan119901119905

119905 (46)


119906 =sin119901119905

119905 V =

tan119901119905

119905 (47)


1 lt 119906(1

cos119901119905)

1(119901+1)

= 119906119901(119901+1)V1(119901+1) (48)





References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










=sin119901119905

sinhminus1119901(tan119901119905)

=sin119901119905

tanhminus1119901(sin119901119905)

(38)




(cosh119901119905)1(119901+1)

lt [

[

sinh119901119905

tanminus1119901(sinh119901119905)

]

]

1119901

ltsinh119901119905

119905 119905 gt 0

(39)


10038161003816100381610038161003816cosh11990111990510038161003816100381610038161003816

119901

minus10038161003816100381610038161003816sinh11990111990510038161003816100381610038161003816

119901

= 1 119901 gt 1 119905 isin R (40)




119901 + 1 lt 119901sin119901119905

119905+tan119901119905

119905 (41)

2 lt (sin119901119905

119905)

119901

+tan119901119905

119905 (42)


119901119905119905






119901 + 1 lt 119901119905

sin119901119905+

119905

tan119901119905 (43)

2 lt (119905

sin119901119905)

119901

+119905

tan119901119905

(44)



119906 =sin119901119905

119905 V =

tan119901119905

119905 (45)





(119905

sin119901119905)

119901

+119905

tan119901119905lt (

sin119901119905

119905)

119901

+tan119901119905

119905 (46)


119906 =sin119901119905

119905 V =

tan119901119905

119905 (47)


1 lt 119906(1

cos119901119905)

1(119901+1)

= 119906119901(119901+1)V1(119901+1) (48)





References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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