For $\alpha>\beta-1>0$, we obtain two sided inequalities for the moment
integral $I(\alpha,\beta)= \int_{\mathbb{R}} |x|^{-\beta}|\sin x|^{\alpha}dx$.
These are then used to give the exact asymptotic behavior of the integral as
$\alpha \to \infty$. The case $I(\alpha,\alpha)$ corresponds to the asymptotics
of Ball's inequality, and $I(\alpha,[\alpha]-1)$ corresponds to a kind of novel
"oscillatory" behavior

Abi-Khuzam, Faruk

Journal of Inequalities and Applications

English

arXiv

Abstract For α > β − 1 > 0 $\alpha>\beta-1>0$ , we obtain two-sided inequalities for the moment integral I ( α , β ) = ∫ R | x | − β | sin x | α d x $I(\alpha,\beta)=\int_{\mathbb{R}}|x|^{-\beta}|\sin x|^{\alpha}\,dx$ . These are then used to give the exact asymptotic behavior of the integral as α → ∞ $\alpha\rightarrow\infty$ . The case I ( α , α ) $I(\alpha,\alpha)$ corresponds to the asymptotics of Ball’s inequality, and I ( α , [ α ] − 1 ) $I(\alpha,[\alpha]-1)$ corresponds to a kind of novel “oscillatory” behavior

Faruk Abi-Khuzam

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Inequalities and asymptotics for some moment integrals

For α> β− 1 > 0 , we obtain two-sided inequalities for the moment integral I(=∫R|x|−β|sinx|αdx. These are then used to give the exact asymptotic behavior of the integral as α→ ∞. The case I(α, α) corresponds to the asymptotics of Ball’s inequality, and I(α, [α] − 1) corresponds to a kind of novel “oscillatory” behavior. © 2017, The Author(s)

Abi-Khuzam, Faruk F.

AUB ScholarWorks (American Univ. of Beirut)

Abi-Khuzam Journal of Inequalities and Applications  (2017) 2017:257 DOI 10.1186/s13660-017-1532-7R E S E A R C H Open AccessInequalities and asymptotics for somemoment integralsFaruk Abi-Khuzam**Correspondence:farukakh@aub.edu.lbDepartment of Mathematics,American University of Beirut,Beirut, LebanonAbstractFor α > β – 1 > 0, we obtain two-sided inequalities for the moment integralI(α,β) =∫R|x|–β | sin x|α dx. These are then used to give the exact asymptotic behaviorof the integral as α → ∞. The case I(α,α) corresponds to the asymptotics of Ball’sinequality, and I(α, [α] – 1) corresponds to a kind of novel “oscillatory” behavior.MSC: Primary 52A20Keywords: Ball’s inequality; asymptotics; cube slicing; moments1 IntroductionBall’s integral inequality [], in connection with cube-slicing in Rn, says that for all s ≥ ,∫ ∞–∞∣∣∣∣sin(πx)πx∣∣∣∣sdx ≤√s, or∫ ∞–∞∣∣∣∣sin xx∣∣∣∣sdx ≤ π√s,with strict inequality except when s = . In particular, it suggests that the integral decayslike √s as s → ∞, and this is made precise by the following asymptotic []:lims→∞√s∫ ∞–∞∣∣∣∣sin(πx)πx∣∣∣∣sdx =√π.Since√π< , the asymptotic result implies the inequality for large values of s. But thereare no known “easy” proofs of the inequality for the full range of values, the main diffi-culty being near small values of s, e.g., between  and  []. The asymptotic result, thoughreasonably tame, presents new difficulties when we consider a more general integral, andthis is circumvented here by the proof of two new inequalities.Our purpose here is to consider a generalization involving the “moment” integralI(α,β) =∫ ∞–∞| sin x|α|x|β dx, α > β –  > .We shall obtain useful upper and lower bounds for this integral and use them to obtain theasymptotic behavior of this integral. In addition, the inequalities obtained are indispens-able in obtaining the asymptotic behavior, especially in the interesting “oscillatory” cases© The Author(s) 2017. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License(http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in anymedium, pro-vided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, andindicate if changes were made.Abi-Khuzam Journal of Inequalities and Applications  (2017) 2017:257 Page 2 of 8I(α, [α]) if α ≥  and I([α],α – ) if α > , where [α] is the greatest integer in α. The oscil-latory behavior makes it impossible to employ the standard methods used in connectionwith Ball’s inequality.We place no restrictions on the indices α and β beyond those necessary to ensure theconvergence of the integral I(α,β). Indeed, the condition β >  implies convergence in aneighborhood of ∞, and near , the inequality | sin x|α|x|β ≤ |x|α–β implies convergence sinceα – β > –.2 Weaker versions of Ball’s inequalityA natural way to deal with Ball’s inequality is to apply the sharp form of the Hausdorff-Young inequality []. This leads to two inequalities for the relevant integral: the first worksfor all s ≥ , but falls short of the required inequality by supplying the larger constant √ein place of√. The second gives a constant smaller than√ but only works for s ≥ .Proposition (a) If s ≥ , then∫ ∞–∞∣∣∣∣sin(πx)πx∣∣∣∣sdx ≤ √s·√( +s – )s–<√es.(b) If s ≥ , q = s , and p is the index conjugate to q, then∫R∣∣∣∣(sinπξπξ)∣∣∣∣sdξ ≤√s(√pp + )q/p<√s.Proof For part (a), let χ = χ(–/,/) be the characteristic function of the interval (–/, /).Then its Fourier transform is given by χ̂ (ξ ) =∫ ∞–∞ χ (x)e–π ixξ dx = sinπξπξ. Applying thesharp Hausdorff-Young inequality [], ‖χ̂‖s ≤ Cp‖χ‖p, where s ≥ , p = s′, the index con-jugate to s, and Cp is given by Cp = p/p(s)–/s, we obtain∫R∣∣∣∣sinπξπξ∣∣∣∣sdξ ≤ (p/ps–/s)s/.It now remains to compute(p/ps–/s)s/ =√s·(ss – ) s–=√s·√( +s – )s–<√es, s ≥ ,and the inequality in (a) follows.To prove part (b), we employ the convolution g = χ ∗ χ of the same characteristic func-tion. A simple computation givesg(x) =⎧⎪⎨⎪⎩ – x  ≤ x ≤  + x – ≤ x ≤  |x| ≥ ⎫⎪⎬⎪⎭.Abi-Khuzam Journal of Inequalities and Applications  (2017) 2017:257 Page 3 of 8Now ĝ(ξ ) = ( sinπξπξ), ‖g‖pp = p+ , and an application of the sharp-Hausdorff-Young inequal-ity gives, for q ≥  and the conjugate index p = q′,∫R∣∣∣∣(sinπξπξ)∣∣∣∣qdξ ≤ (p/pq–/q)q/(p + )q/p= q–/(√pp + )q/p<√pp + ·√q.Since √pp+ < , we obtain the inequality∫ ∞–∞ | sin(πx)πx |s dx <√s for all s ≥ . 3 Main resultsIn this section we consider the question of obtaining upper and lower bounds for the moregeneral integral, namely∫ ∞–∞| sin x|α|x|β dx. Those bounds are then used to obtain the preciseasymptotic behavior of the integral as α → ∞. In addition, the bounds make it possible toemploy discontinuous functions such as [α] in place of β , and then the asymptotic resultalso captures the precise oscillations in the values of the integral, as α → ∞.Theorem  Suppose α > β –  > , and putI(α,β) =∫R| sin t|α|t|β dt = ∫ ∞| sin t|α|t|β dt,andφ(α,β) =αα–β+ (α + )( α–β+ + α + ),where  is the gamma-function. Then(α) α–β+(α – β + )φ(α,β) ≤ I(α,β) ≤(α) α–β+(α – β + ){ +β – }.In particular, if β = α, then√α√α(α + )(α +  )√π ≤ I(α,α) ≤(√α)√π{ +α – }.Proof We need first the following double inequality: –x≤ sin xx≤ e– x ,  ≤ x ≤ π .The left-hand inequality is easily proved by calculus. It will be used with  ≤ x ≤ √.For the right-hand inequality, since  ≤ x ≤ π , we may use the inequality between thegeometric and arithmetic mean of positive numbers to obtainn∏k=( –xπk)≤( –xπnn∑k=k)n≤ exp(–xπn∑k=k).Abi-Khuzam Journal of Inequalities and Applications  (2017) 2017:257 Page 4 of 8Letting n → ∞ and recalling the product representation of the sine function and∑∞k=k =π , we obtain the second inequality. The next step is to compare the full in-tegral in the theorem to an integral over the interval [,√], or over [,π ].∫ √| sin t|α|t|β dt ≤∫ π| sin t|α|t|β dt ≤∫ ∞| sin t|α|t|β dt=∫ π| sin t|α|t|β dt +∞∑k=∫ (k+)πkπ| sin t|α|t|β dt=∫ π| sin t|α|t|β dt +∞∑k=∫ π| sin t|α|t + kπ |β dt≤{ +∞∑k=(k + )β}∫ π| sin t|α|t|β dt≤{ +β – }∫ πsinα ttβdt.Using the above inequalities for sin xx ,∫ √tα–β( –t)αdt ≤∫ πsinα ttβdt ≤∫ πtα–β exp(–αt)dt.Simple substitutions to change variables bring this double inequality to the formα–β+∫ xα–β– ( – x)α dx ≤∫ πsinα ttβdt ≤ (α) α–β+∫ παxα–β– e–x dx.If we extend the right most integral to [,∞), and then express both sides through thegamma function, we arrive atα–β+( α–β+ )(α + )( α–β+ + α + )≤∫ πsinα ttβdt ≤ (α) α–β+(α – β + ).This gives the first inequalities for I(α,β), and so, the inequalities for I(α,α). Corollary  Let I(α,β) be the integral in the theorem.(a) If α – β = c is constant, while α → ∞, thenlimα→∞αc+ I(α,β) = c+ (c + ), c > –.In particular, the asymptotic for the integral in Ball’s inequality islimα→∞√αI(α,α) =√π .(b) If α – β = c, and c remains bounded as α → ∞, thenI(α,β) (α) α–β+(α – β + ), α → ∞.Abi-Khuzam Journal of Inequalities and Applications  (2017) 2017:257 Page 5 of 8In particular,I(α, [α])(α) α–[α]+(α – [α] + ), α → ∞.Proof (a) In the very special case where β = α, Stirling’s formula givesφ(α,α) =√α(α + )(α +  )∼αα+e/(α + /)α+=e/( + α )α+∼ , α → ∞.From this, the case where α – β = c, a constant, is handled similarly:φ(α,β) =αc+ (α + )( c+ + α + )∼αα+c+ + e–α(α + c+ )α+ c+ + e–(α+ c+ )=e c+( + c+α )α+ c+ +∼ , α → ∞.(c) If α – β = c > –, and c is only bounded, then Stirling’s formula followed by the in-equality ( + c+α )α ≤ e c+ , giveslim infα→∞ φ(α,β) = lim infα→∞αc+ (α + )( c+ + α + )= lim infα→∞e c+( + c+α )α+ c+ +≥ lim infα→∞( + c+α )c+ += .So thatlim infα→∞[(α) α–β+(α – β + )]–I(α,β) ≥ .The corresponding lim sup being clearly ≤ , we obtainI(α,β) (α) α–β+(α – β + ), α – β bounded,β → ∞. 4 ConclusionFor α > β – > , we have obtained two-sided inequalities for the moment integral I(α,β) =∫R|x|–β | sin x|α dx and used them to predict and then prove the exact asymptotic behaviorof the integral as α → ∞. In particular, we showed that the asymptotic behavior I(α, [α]–)corresponds to the oscillations of ( α–[α]+ ) between its extreme values. We would like toend by a generalization of the asymptotic result for a class of infinite products. Let g be afunction having an infinite product representation of the formg(t) =∞∏n=( –ttn),Abi-Khuzam Journal of Inequalities and Applications  (2017) 2017:257 Page 6 of 8where tn >  and c =∑∞n= t–n < ∞. We are interested in investigating limp→∞∫ ∞|g(t)|ptβ dt,where  ≤ β < . Two examples of such a function aref (x) =∫ cos(xt)h(t) dt, J(x) =π∫ cos xt√ – tdt.[Where in the first integral h is continuously differentiable, h(t) > , h′(t) < , and h′′(t) < for  ≤ t ≤ .] The first function f was considered in [] in connection with maximalmeasures of sections of the n-cube. The second is the Bessel function of order .In studying the asymptotics of∫ ∞|g(t)|ptβ dt, it appears instructive to consider first thecase where β = .Proposition  If g is the function defined above, then we have the double inequality – ct ≤ ∣∣g(t)∣∣ ≤ e–ct , c =∞∑k=t–k ,where the left inequality is used when  ≤ t ≤ √c , and the right inequality when  ≤ t ≤ t.Furthermore,limp→∞√p∫ ∞–∞∣∣g(t)∣∣p dt =√πc.Proof In general, if  < ak < , k = , , . . . , n, then – (a + a + · · · + an) ≤n∏k=( – ak) ≤( –nn∑k=ak)n.The inequality on the left is proved by induction. The inequality on the right is the AGM(arithmetic-geometric mean) inequality. Taking ak = ( ttk ), with  ≤ t ≤ t, gives( – tn∑k=t–k)≤n∏k=( –ttk)≤( –tnn∑k=t–k)n≤ exp(–tn∑k=t–k),and passing to the limit as n → ∞, we arrive at the required inequalities.The left-hand inequality gives∫ ∞∣∣g(t)∣∣p dt ≥∫ √c( – ct)p dt =√c∫ ( – x)px–/ dx =√c(p + )√π(p +  ).By Stirling’s formula we obtainlim infp→∞√p∫ ∞–∞∣∣g(t)∣∣p dt ≥√πc,which suggests that the order of decay of the integral is √p , and this leads naturally toa consideration of √p ∫ ∞ |g(t)|p dt =∫ ∞ |g( t√p )|p dt. Now a substitution in the two-sidedAbi-Khuzam Journal of Inequalities and Applications  (2017) 2017:257 Page 7 of 8inequalities above gives( – ctp)p≤∣∣∣∣g(t√p)∣∣∣∣p≤ e–ct ,where the left-hand inequality holds true for  ≤ t ≤√p√c , and the right-hand inequal-ity holds true for  ≤ t ≤ t√p. It now becomes possible to use, exactly as done in[], Lebesgue’s dominated convergence theorem to conclude that actually limp→∞√p ×∫ ∞–∞ |g(t)|p dt =√πc . Our final result handles the general case where  < β < .Theorem  If g is the function considered above, and  < β < , thenlimp→∞ p–β∫ ∞|g(t)|ptβdt =( –β )√c–β.Proof In following the same approach as in the proof of Proposition  above, we need toknow beforehand the expected rate of decay. Thus, using one of the inequalities in Propo-sition , we obtain∫ ∞|g(t)|ptβdt ≥∫ √c( – ct)pt–β dt =√c–β∫ ( – x)px–+β dx=√c–β(p + )( –β )( –β + p + ),leading to a sharp lower asymptotic, namelylim infp→∞ p–β∫ ∞|g(t)|ptβdt ≥ (–β )√c–β.Once again this suggests that the expected decay is like pβ– . So we make the substitutiont = ( – β)–β p–/x–β and find thatp–β∫ ∞|g(t)|ptβdt =∫ ∞∣∣∣∣g(( – β)–β x–β√p)∣∣∣∣pdx.The inequalities( – c( – β)–β x–βp)p≤∣∣∣∣g(( – β)–β x–β√p)∣∣∣∣p≤ exp(–c( – β) –β x –β )make it possible to use Lebesgue’s theorem, and we arrive at the asymptoticlimp→∞ p–β∫ ∞|g(t)|ptβdt =( –β )√c–β. Competing interestsThe author declares that he has no competing interests.Abi-Khuzam Journal of Inequalities and Applications  (2017) 2017:257 Page 8 of 8Authors’ contributionsThe author declares that there are no other contributors to this article. The author read and approved the finalmanuscript.Publisher’s NoteSpringer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.Received: 28 April 2017 Accepted: 22 September 2017References1. Ball, K: Cube slicing in Rn . Proc. Am. Math. Soc. 97(3), 465-473 (1986)2. Nazarov, FL, Podkorytov, AN: Ball, Haagerup, and distribution functions, complex analysis, operators, and related topics.In: Oper. Theory Adv. Appl., vol. 113, pp. 247-267. Birkhäuser, Basel (2000)3. Beckner, W: Inequalities in Fourier analysis. Ann. Math. (2) 102(1), 159-182 (1975)4. Konig, H, Koldobsky, A: On the maximal measure of sections of the n-cube. Contemp. Math. 599, 123-155 (2013)

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Abi-Khuzam Journal of Inequalities and Applications  (2017) 2017:257 DOI 10.1186/s13660-017-1532-7RESEARCH Open AccessInequalities and asymptotics for somemoment integralsFaruk Abi-Khuzam**Correspondence:farukakh@aub.edu.lbDepartment of Mathematics,American University of Beirut,Beirut, LebanonAbstractFor α > β – 1 > 0, we obtain two-sided inequalities for the moment integralI(α,β) =∫R|x|–β | sin x|α dx. These are then used to give the exact asymptotic behaviorof the integral as α → ∞. The case I(α,α) corresponds to the asymptotics of Ball’sinequality, and I(α, [α] – 1) corresponds to a kind of novel “oscillatory” behavior.MSC: Primary 52A20Keywords: Ball’s inequality; asymptotics; cube slicing; moments1 IntroductionBall’s integral inequality [], in connection with cube-slicing in Rn, says that for all s≥ ,∫ ∞–∞∣∣∣∣sin(πx)πx∣∣∣∣sdx≤√s , or∫ ∞–∞∣∣∣∣sinxx∣∣∣∣sdx≤ π√s ,with strict inequality except when s = . In particular, it suggests that the integral decayslike √s as s→ ∞, and this is made precise by the following asymptotic []:lims→∞√s∫ ∞–∞∣∣∣∣sin(πx)πx∣∣∣∣sdx =√π.Since√π< , the asymptotic result implies the inequality for large values of s. But thereare no known “easy” proofs of the inequality for the full range of values, the main diﬃ-culty being near small values of s, e.g., between  and  []. The asymptotic result, thoughreasonably tame, presents new diﬃculties when we consider a more general integral, andthis is circumvented here by the proof of two new inequalities.Our purpose here is to consider a generalization involving the “moment” integralI(α,β) =∫ ∞–∞| sinx|α|x|β dx, α > β –  > .We shall obtain useful upper and lower bounds for this integral and use them to obtain theasymptotic behavior of this integral. In addition, the inequalities obtained are indispens-able in obtaining the asymptotic behavior, especially in the interesting “oscillatory” cases© The Author(s) 2017. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License(http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in anymedium, pro-vided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, andindicate if changes were made.Abi-Khuzam Journal of Inequalities and Applications  (2017) 2017:257 Page 2 of 8I(α, [α]) if α ≥  and I([α],α – ) if α > , where [α] is the greatest integer in α. The oscil-latory behavior makes it impossible to employ the standard methods used in connectionwith Ball’s inequality.We place no restrictions on the indices α and β beyond those necessary to ensure theconvergence of the integral I(α,β). Indeed, the condition β >  implies convergence in aneighborhood of ∞, and near , the inequality | sinx|α|x|β ≤ |x|α–β implies convergence sinceα – β > –.2 Weaker versions of Ball’s inequalityA natural way to deal with Ball’s inequality is to apply the sharp form of the Hausdorﬀ-Young inequality []. This leads to two inequalities for the relevant integral: the ﬁrst worksfor all s ≥ , but falls short of the required inequality by supplying the larger constant √ein place of√. The second gives a constant smaller than√ but only works for s≥ .Proposition (a) If s≥ , then∫ ∞–∞∣∣∣∣sin(πx)πx∣∣∣∣sdx≤ √s ·√( + s – )s–<√es .(b) If s≥ , q = s , and p is the index conjugate to q, then∫R∣∣∣∣(sinπξπξ)∣∣∣∣sdξ ≤√s(√pp + )q/p<√s .Proof For part (a), let χ = χ(–/,/) be the characteristic function of the interval (–/, /).Then its Fourier transform is given by χˆ (ξ ) =∫ ∞–∞ χ (x)e–π ixξ dx =sinπξπξ. Applying thesharp Hausdorﬀ-Young inequality [], ‖χˆ‖s ≤ Cp‖χ‖p, where s ≥ , p = s′, the index con-jugate to s, and Cp is given by Cp = p/p(s)–/s, we obtain∫R∣∣∣∣sinπξπξ∣∣∣∣sdξ ≤ (p/ps–/s)s/.It now remains to compute(p/ps–/s)s/ = √s ·( ss – ) s–= √s ·√( + s – )s–<√es , s≥ ,and the inequality in (a) follows.To prove part (b), we employ the convolution g = χ ∗ χ of the same characteristic func-tion. A simple computation givesg(x) =⎧⎪⎨⎪⎩ – x ≤ x≤  + x –≤ x≤  |x| ≥ ⎫⎪⎬⎪⎭.Abi-Khuzam Journal of Inequalities and Applications  (2017) 2017:257 Page 3 of 8Now gˆ(ξ ) = ( sinπξπξ), ‖g‖pp = p+ , and an application of the sharp-Hausdorﬀ-Young inequal-ity gives, for q ≥  and the conjugate index p = q′,∫R∣∣∣∣(sinπξπξ)∣∣∣∣qdξ ≤ (p/pq–/q)q/( p + )q/p= q–/(√pp + )q/p< √pp +  ·√q .Since √pp+ < , we obtain the inequality∫ ∞–∞ | sin(πx)πx |s dx <√s for all s≥ . 3 Main resultsIn this section we consider the question of obtaining upper and lower bounds for themoregeneral integral, namely∫ ∞–∞| sinx|α|x|β dx. Those bounds are then used to obtain the preciseasymptotic behavior of the integral as α → ∞. In addition, the bounds make it possible toemploy discontinuous functions such as [α] in place of β , and then the asymptotic resultalso captures the precise oscillations in the values of the integral, as α → ∞.Theorem  Suppose α > β –  > , and putI(α,β) =∫R| sin t|α|t|β dt = ∫ ∞| sin t|α|t|β dt,andφ(α,β) = αα–β+ (α + )( α–β+ + α + ),where  is the gamma-function. Then(α) α–β+(α – β + )φ(α,β)≤ I(α,β)≤(α) α–β+(α – β + ){ + β – }.In particular, if β = α, then√α√α(α + )(α +  )√π ≤ I(α,α)≤(√α)√π{ + α – }.Proof We need ﬁrst the following double inequality: – x ≤sinxx ≤ e– x , ≤ x≤ π .The left-hand inequality is easily proved by calculus. It will be used with  ≤ x ≤ √.For the right-hand inequality, since  ≤ x ≤ π , we may use the inequality between thegeometric and arithmetic mean of positive numbers to obtainn∏k=( – xπk)≤( – xπnn∑k=k)n≤ exp(– xπn∑k=k).Abi-Khuzam Journal of Inequalities and Applications  (2017) 2017:257 Page 4 of 8Letting n → ∞ and recalling the product representation of the sine function and∑∞k=k =π , we obtain the second inequality. The next step is to compare the full in-tegral in the theorem to an integral over the interval [,√], or over [,π ].∫ √| sin t|α|t|β dt ≤∫ π| sin t|α|t|β dt ≤∫ ∞| sin t|α|t|β dt=∫ π| sin t|α|t|β dt +∞∑k=∫ (k+)πkπ| sin t|α|t|β dt=∫ π| sin t|α|t|β dt +∞∑k=∫ π| sin t|α|t + kπ |β dt≤{ +∞∑k=(k + )β}∫ π| sin t|α|t|β dt≤{ + β – }∫ πsinα ttβ dt.Using the above inequalities for sinxx ,∫ √tα–β( – t)αdt ≤∫ πsinα ttβ dt ≤∫ πtα–β exp(–αt)dt.Simple substitutions to change variables bring this double inequality to the formα–β+∫ xα–β– ( – x)α dx≤∫ πsinα ttβ dt ≤(α) α–β+ ∫ παxα–β– e–x dx.If we extend the right most integral to [,∞), and then express both sides through thegamma function, we arrive atα–β+( α–β+ )(α + )( α–β+ + α + )≤∫ πsinα ttβ dt ≤(α) α–β+(α – β + ).This gives the ﬁrst inequalities for I(α,β), and so, the inequalities for I(α,α). Corollary  Let I(α,β) be the integral in the theorem.(a) If α – β = c is constant, while α → ∞, thenlimα→∞αc+ I(α,β) =  c+ ( c + ), c > –.In particular, the asymptotic for the integral in Ball’s inequality islimα→∞√αI(α,α) =√π .(b) If α – β = c, and c remains bounded as α → ∞, thenI(α,β)(α) α–β+(α – β + ), α → ∞.Abi-Khuzam Journal of Inequalities and Applications  (2017) 2017:257 Page 5 of 8In particular,I(α, [α])(α) α–[α]+(α – [α] + ), α → ∞.Proof (a) In the very special case where β = α, Stirling’s formula givesφ(α,α) =√α(α + )(α +  )∼αα+e/(α + /)α+ =e/( + α )α+∼ , α → ∞.From this, the case where α – β = c, a constant, is handled similarly:φ(α,β) = αc+ (α + )( c+ + α + )∼αα+c+ + e–α(α + c+ )α+c+ + e–(α+ c+ )= ec+( + c+α )α+c+ +∼ , α → ∞.(c) If α – β = c > –, and c is only bounded, then Stirling’s formula followed by the in-equality ( + c+α )α ≤ ec+ , giveslim infα→∞ φ(α,β) = lim infα→∞αc+ (α + )( c+ + α + )= lim infα→∞e c+( + c+α )α+c+ +≥ lim infα→∞( + c+α )c+ += .So thatlim infα→∞[(α) α–β+(α – β + )]–I(α,β)≥ .The corresponding lim sup being clearly ≤ , we obtainI(α,β)(α) α–β+(α – β + ), α – β bounded,β → ∞. 4 ConclusionFor α > β– > , we have obtained two-sided inequalities for themoment integral I(α,β) =∫R|x|–β | sinx|α dx and used them to predict and then prove the exact asymptotic behaviorof the integral asα → ∞. In particular, we showed that the asymptotic behavior I(α, [α]–)corresponds to the oscillations of ( α–[α]+ ) between its extreme values. We would like toend by a generalization of the asymptotic result for a class of inﬁnite products. Let g be afunction having an inﬁnite product representation of the formg(t) =∞∏n=( – ttn),Abi-Khuzam Journal of Inequalities and Applications  (2017) 2017:257 Page 6 of 8where tn >  and c =∑∞n= t–n <∞. We are interested in investigating limp→∞∫ ∞|g(t)|ptβ dt,where ≤ β < . Two examples of such a function aref (x) =∫ cos(xt)h(t)dt, J(x) =π∫ cosxt√ – tdt.[Where in the ﬁrst integral h is continuously diﬀerentiable, h(t) > , h′(t) < , and h′′(t) < for  ≤ t ≤ .] The ﬁrst function f was considered in [] in connection with maximalmeasures of sections of the n-cube. The second is the Bessel function of order .In studying the asymptotics of∫ ∞|g(t)|ptβ dt, it appears instructive to consider ﬁrst thecase where β = .Proposition  If g is the function deﬁned above, then we have the double inequality – ct ≤ ∣∣g(t)∣∣ ≤ e–ct , c =∞∑k=t–k ,where the left inequality is used when ≤ t ≤ √c , and the right inequality when ≤ t ≤ t.Furthermore,limp→∞√p∫ ∞–∞∣∣g(t)∣∣p dt =√πc .Proof In general, if  < ak < , k = , , . . . ,n, then – (a + a + · · · + an)≤n∏k=( – ak)≤( – nn∑k=ak)n.The inequality on the left is proved by induction. The inequality on the right is the AGM(arithmetic-geometric mean) inequality. Taking ak = ( ttk ), with ≤ t ≤ t, gives( – tn∑k=t–k)≤n∏k=( – ttk)≤( – tnn∑k=t–k)n≤ exp(–tn∑k=t–k),and passing to the limit as n→ ∞, we arrive at the required inequalities.The left-hand inequality gives∫ ∞∣∣g(t)∣∣p dt ≥∫ √c( – ct)p dt = √c∫ ( – x)px–/ dx = √c(p + )√π(p +  ).By Stirling’s formula we obtainlim infp→∞√p∫ ∞–∞∣∣g(t)∣∣p dt ≥√πc ,which suggests that the order of decay of the integral is √p , and this leads naturally toa consideration of √p ∫ ∞ |g(t)|p dt =∫ ∞ |g( t√p )|p dt. Now a substitution in the two-sidedAbi-Khuzam Journal of Inequalities and Applications  (2017) 2017:257 Page 7 of 8inequalities above gives( – c tp)p≤∣∣∣∣g( t√p)∣∣∣∣p≤ e–ct ,where the left-hand inequality holds true for  ≤ t ≤√p√c , and the right-hand inequal-ity holds true for  ≤ t ≤ t√p. It now becomes possible to use, exactly as done in[], Lebesgue’s dominated convergence theorem to conclude that actually limp→∞√p×∫ ∞–∞ |g(t)|p dt =√πc . Our ﬁnal result handles the general case where  < β < .Theorem  If g is the function considered above, and  < β < , thenlimp→∞p–β∫ ∞|g(t)|ptβ dt =( –β )√c–β.Proof In following the same approach as in the proof of Proposition  above, we need toknow beforehand the expected rate of decay. Thus, using one of the inequalities in Propo-sition , we obtain∫ ∞|g(t)|ptβ dt ≥∫ √c( – ct)pt–β dt = √c–β∫ ( – x)px–+β dx= √c–β(p + )( –β )( –β + p + ),leading to a sharp lower asymptotic, namelylim infp→∞ p–β∫ ∞|g(t)|ptβ dt ≥( –β )√c–β.Once again this suggests that the expected decay is like p β– . So we make the substitutiont = ( – β)–β p–/x–β and ﬁnd thatp–β∫ ∞|g(t)|ptβ dt =∫ ∞∣∣∣∣g( ( – β)–β x–β√p)∣∣∣∣pdx.The inequalities( – c ( – β)–β x–βp)p≤∣∣∣∣g( ( – β)–β x–β√p)∣∣∣∣p≤ exp(–c( – β) –β x –β )make it possible to use Lebesgue’s theorem, and we arrive at the asymptoticlimp→∞p–β∫ ∞|g(t)|ptβ dt =( –β )√c–β. Competing interestsThe author declares that he has no competing interests.Abi-Khuzam Journal of Inequalities and Applications  (2017) 2017:257 Page 8 of 8Authors’ contributionsThe author declares that there are no other contributors to this article. The author read and approved the ﬁnalmanuscript.Publisher’s NoteSpringer Nature remains neutral with regard to jurisdictional claims in published maps and institutional aﬃliations.Received: 28 April 2017 Accepted: 22 September 2017References1. Ball, K: Cube slicing in Rn . Proc. Am. Math. Soc. 97(3), 465-473 (1986)2. Nazarov, FL, Podkorytov, AN: Ball, Haagerup, and distribution functions, complex analysis, operators, and related topics.In: Oper. Theory Adv. Appl., vol. 113, pp. 247-267. Birkhäuser, Basel (2000)3. Beckner, W: Inequalities in Fourier analysis. Ann. Math. (2) 102(1), 159-182 (1975)4. Konig, H, Koldobsky, A: On the maximal measure of sections of the n-cube. Contemp. Math. 599, 123-155 (2013)

Inequalities and Asymptotics for some Moment Integrals

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