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based on infinite product expansions and inequalities on series and the riemann s zeta function , daurizio ( @xcite ) proved the following inequality : @xmath0 where @xmath1 . using an elementary approach , sndor ( @xcite ) offered an alternative proof of by employing trigonometric inequalities and an auxiliary function . in the same paper , sndor also provided the converse to : @xmath2 where @xmath1 . in addition , sndor found the following analogous inequality holds true for the case of sine functions :    [ thm : daurizio - sandor ineq original ] the two double inequalities @xmath3 and @xmath4 hold for any @xmath1 .    throughout this paper , we denote @xmath5 and @xmath6 by @xmath7 and @xmath8 , respectively : @xmath9    our aim is to generalize the daurizio - sndor inequalities for the case of @xmath10 and @xmath11 as follows :    [ thm : daurizio - sandor ineq generalized ] let @xmath12 . then the two double inequalities @xmath13 and @xmath14 hold for @xmath15 . in particular , the double inequality remains true when @xmath16 while the double inequality is reversed when @xmath16 . the remainder of this paper is organized as follows . section  [ sec : proof of the main results ] is devoted to the proof of theorem  [ thm : daurizio - sandor ineq generalized ] and an alternative proof of theorem  [ thm : daurizio - sandor ineq original ] . in section  [ sec : daurizio - sandor inequalities for hyperbolic functions ] , we establish analogue of theorem  [ thm : daurizio - sandor ineq generalized ] for hyperbolic functions . as an application of theorem  [ thm : daurizio - sandor ineq generalized ] , we apply in section  [ sec : application ] inequality to the chebyshev polynomials of the second kind and establish a trigonometric inequality . at first we will prove the following lemma . the lemma provides expressions of the higher - order derivative @xmath17 involving @xmath18 @xmath19 , which are helpful in proving theorem  [ thm : daurizio - sandor ineq generalized ] . we note that the sign of @xmath17 plays a crucial role in proving theorem  [ thm : daurizio - sandor ineq generalized ] . [ lem : lots of f ] let @xmath12 and @xmath20 . then when @xmath21 and @xmath22 , we have    @xmath23 : :    @xmath24 @xmath25 : :    @xmath26    +    in particular , @xmath27 : :    when @xmath28 , @xmath29 @xmath30 : :    when @xmath31 , @xmath32 @xmath33 : :    for @xmath34 and    @xmath35 ,    @xmath36    @xmath23 , @xmath25 and @xmath33 follows directly from calculations using elementary calculus . in particular , trigonometric addition formulas are used in proving @xmath23 and @xmath25 . to prove , we claim @xmath37 indeed , we rewrite @xmath38 on the other hand , making use of euler s formula @xmath39 leads to an alternative expression of the left - hand side of : @xmath40 where @xmath41 is the real part of @xmath42 and @xmath43 . now it suffices to show @xmath44 using in @xmath25 , this can be achieved by straightforward calculations . thus @xmath27 is true . the proof of @xmath30 is similar , and we omit the details . we complete the proof of lemma  [ lem : lots of f ] . we provide here an alternative proof of the two double inequalities in theorem  [ thm : daurizio - sandor ineq original ] .    to this end , we show that for @xmath1 , @xmath45 is strictly increasing while @xmath46 is strictly decreasing . these lead to the desired inequalities since it is easy to see that @xmath47 to see @xmath48 is strictly increasing , we employ in lemma  [ lem : lots of f ] to obtain @xmath49 as @xmath50 , it follows that @xmath51 . we are led to @xmath52 or @xmath53 since @xmath54 . this that shows @xmath48 is strictly increasing .    by using in lemma  [ lem : lots of f ] , we have @xmath55 from which we infer that @xmath56 since @xmath57 by @xmath33 of lemma  [ lem : lots of f ] . then @xmath58 together with the fact @xmath59 from @xmath33 of lemma  [ lem : lots of f ] yields @xmath60 or @xmath61 . thus we have shown that @xmath62 is strictly decreasing . this completes the proof of the theorem . we are now in the position to give the proof of theorem  [ thm : daurizio - sandor ineq generalized ] . the proof of the case when @xmath16 has been given in theorem  [ thm : daurizio - sandor ineq generalized ] . for @xmath63 , we prove the desired inequalities by showing that @xmath64 for @xmath34 . due to @xmath23 of lemma  [ lem : lots of f ] , we see that @xmath65 for @xmath63 . instead of employing @xmath25 of lemma  [ lem : lots of f ] , we use and in lemma  [ lem : lots of f ] to conclude that @xmath66 . thus we have for @xmath34 , @xmath67 because of the first vanishing limit in @xmath33 of lemma  [ lem : lots of f ] , it follows that @xmath68 which , together with the fact that the second limit in @xmath33 of lemma  [ lem : lots of f ] vanishes , implies that @xmath69 or @xmath64 for @xmath34 . it remains to find the following limits : @xmath70 we immediately have @xmath71 and @xmath72 the proof is completed . in this section , we show an analogue of theorem  [ thm : daurizio - sandor ineq generalized ] for the case of hyperbolic functions holds true . let @xmath73 following the same arguments for proving lemma  [ lem : lots of f ] , it can be shown that lemma  [ lem : lots of f ] with @xmath74 , @xmath75 and @xmath18 @xmath19 replaced by @xmath76 , @xmath77 and @xmath78 @xmath19 respectively , remains true . it follows that we can prove @xmath79 for @xmath34 as in the proof of theorem  [ thm : daurizio - sandor ineq generalized ] . it remains to calculate the following limits : @xmath80 thus , we have the following analogue of theorem  [ thm : daurizio - sandor ineq generalized ] for @xmath76 and @xmath77 . [ thm : daurizio - sandor ineq generalized hyperbolic fcn ] let @xmath12 . then the two double inequalities @xmath81 and @xmath82 hold for @xmath15 . in particular , the double inequality is reversed when @xmath16 while the double inequality remains true when @xmath16 . the first few chebyshev polynomials of the second kind @xmath83 @xmath84 are ( @xcite ) @xmath85 in this section , we apply theorem  [ thm : daurizio - sandor ineq generalized ] to @xmath83 with @xmath86 . by means of the formula @xmath87 , we obtain the following corollary . _ acknowledgements_. the authors wish to express sincere gratitude to tom mollee for his careful reading of the manuscript and valuable suggestions to improve the readability of the paper . thanks are also due to chiun - chuan chen and mach nguyet minh for the fruitful discussions . the authors are grateful to the anonymous referee for many helpful comments and valuable suggestions on this paper . m. abramowitz and i. a. stegun ( eds ) handbook of mathematical functions with formulas , graphs , and mathematical tables national bureau of standards , applied mathematics series * 55 * , 9th printing washington 1970
in this work , we generalize the daurizio - sndor inequalities ( @xcite ) using an elementary approach . in particular , our approach provides an alternative proof of the daurizio - sndor inequalities . 
 moreover , as an immediate consequence of the generalized daurizio - sndor inequalities , we establish the daurizio - sndor - type inequalities for hyperbolic functions .