Jensen inequality for integrals

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11: 6.8 Inequalities

§6.8 Inequalities

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6.8.1

\frac{1}{2}\ln\left(1+\frac{2}{x}\right)<e^{x}E_{1}\left(x\right)<\ln\left(1+% \frac{1}{x}\right),

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Symbols:: $\mathrm{e}$ : base of natural logarithm, $E_{1}\left(\NVar{z}\right)$ : exponential integral, $\ln\NVar{z}$ : principal branch of logarithm function and $x$ : real variable
A&S Ref:: 5.1.20
Referenced by:: §6.8
Permalink:: http://dlmf.nist.gov/6.8.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §6.8 and Ch.6

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6.8.2

\frac{x}{x+1}<xe^{x}E_{1}\left(x\right)<\frac{x+1}{x+2},

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Symbols:: $\mathrm{e}$ : base of natural logarithm, $E_{1}\left(\NVar{z}\right)$ : exponential integral and $x$ : real variable
Referenced by:: §6.8
Permalink:: http://dlmf.nist.gov/6.8.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §6.8 and Ch.6

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6.8.3

\frac{x(x+3)}{x^{2}+4x+2}<xe^{x}E_{1}\left(x\right)<\frac{x^{2}+5x+2}{x^{2}+6x% +6}.

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Symbols:: $\mathrm{e}$ : base of natural logarithm, $E_{1}\left(\NVar{z}\right)$ : exponential integral and $x$ : real variable
Referenced by:: §6.8
Permalink:: http://dlmf.nist.gov/6.8.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §6.8 and Ch.6

12: Bibliography Q

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F. Qi and J. Mei (1999) Some inequalities of the incomplete gamma and related functions. Z. Anal. Anwendungen 18 (3), pp. 793–799.

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External Links:: ISSN 0232-2064, FullText, MathReview (Richard B. Paris), ZentralBlatt
Referenced by:: §8.10.
Encodings:: BibTeX

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F. Qi (2008) A new lower bound in the second Kershaw’s double inequality. J. Comput. Appl. Math. 214 (2), pp. 610–616.

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External Links:: ISSN 0377-0427, Document, FullText, MathReview (Mehdi Hassani), ZentralBlatt
Referenced by:: §5.6(i), §5.6(i).
Encodings:: BibTeX

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S.-L. Qiu and M. K. Vamanamurthy (1996) Sharp estimates for complete elliptic integrals. SIAM J. Math. Anal. 27 (3), pp. 823–834.

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External Links:: ISSN 0036-1410, Document, MathReview (G. D. Anderson), ZentralBlatt
Referenced by:: §19.9(i), §19.9(i).
Encodings:: BibTeX

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W.-Y. Qiu and R. Wong (2000) Uniform asymptotic expansions of a double integral: Coalescence of two stationary points. Proc. Roy. Soc. London Ser. A 456, pp. 407–431.

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External Links:: ISSN 1364-5021, Document, MathReview (F. Ursell), ZentralBlatt
Referenced by:: §2.4(vi).
Encodings:: BibTeX

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13: 7.8 Inequalities

§7.8 Inequalities

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7.8.7

\frac{\sinh x^{2}}{x}<{\mathrm{e}}^{x^{2}}F\left(x\right)=\int_{0}^{x}{\mathrm% {e}}^{t^{2}}\,\mathrm{d}t<\frac{{\mathrm{e}}^{x^{2}}-1}{x},

x>0

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Symbols:: $F\left(\NVar{z}\right)$ : Dawson’s integral, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\sinh\NVar{z}$ : hyperbolic sine function, $\int$ : integral and $x$ : real variable
Referenced by:: Erratum (V1.1.5) for Additions
Permalink:: http://dlmf.nist.gov/7.8.E7
Encodings:: pMML, png, TeX
Addition (effective with 1.1.5):: A new inequality was added.
See also:: Annotations for §7.8 and Ch.7

►The function

F\left(x\right)/\sqrt{1-{\mathrm{e}}^{-2x^{2}}}

is strictly decreasing for

x>0

. For these and similar results for Dawson’s integral

F\left(x\right)

see Janssen (2021). ►

7.8.8

\operatorname{erf}x<\sqrt{1-{\mathrm{e}}^{-4x^{2}/\pi}},

x>0

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Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{erf}\NVar{z}$ : error function, $\mathrm{e}$ : base of natural logarithm and $x$ : real variable
Source:: Pólya (1949, (1.5))
Referenced by:: §7.8, Erratum (V1.0.17) for Equation (7.8.8)
Permalink:: http://dlmf.nist.gov/7.8.E8
Encodings:: pMML, png, TeX
Addition (effective with 1.0.17):: This inequality was added. It is Pólya (1949, (1.5)). Note that it is a special case of (8.10.11).
Suggested by Roberto Iacono
See also:: Annotations for §7.8 and Ch.7

15: 19.9 Inequalities

§19.9 Inequalities

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§19.9(i) Complete Integrals

… ►Other inequalities are: … ►

§19.9(ii) Incomplete Integrals

… ►Simple inequalities for incomplete integrals follow directly from the defining integrals (§19.2(ii)) together with (19.6.12): …

16: Bibliography F

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J. Faraut (1982) Un théorème de Paley-Wiener pour la transformation de Fourier sur un espace riemannien symétrique de rang un. J. Funct. Anal. 49 (2), pp. 230–268.

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External Links:: ISSN 0022-1236, Document, Link, MathReview (Mogens Flensted-Jensen), ZentralBlatt
Referenced by:: §18.39(i).
Encodings:: BibTeX

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S. Fempl (1960) Sur certaines sommes des intégral-cosinus. Bull. Soc. Math. Phys. Serbie 12, pp. 13–20 (French).

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External Links:: MathReview (L. Carlitz), ZentralBlatt
Referenced by:: §6.15.
Encodings:: BibTeX

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H. E. Fettis (1970) On the reciprocal modulus relation for elliptic integrals. SIAM J. Math. Anal. 1 (4), pp. 524–526.

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External Links:: Document, MathReview (B. C. Carlson), ZentralBlatt
Referenced by:: §19.7(i).
Encodings:: BibTeX

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C. H. Franke (1965) Numerical evaluation of the elliptic integral of the third kind. Math. Comp. 19 (91), pp. 494–496.

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External Links:: FullText, MathReview, ZentralBlatt
Referenced by:: §19.36(iv).
Encodings:: BibTeX

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T. Fukushima (2012) Series expansions of symmetric elliptic integrals. Math. Comp. 81 (278), pp. 957–990.

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External Links:: Document, ZentralBlatt
Referenced by:: §20.7(ii), §20.7(ii).
Encodings:: BibTeX

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17: 8.10 Inequalities

§8.10 Inequalities

… ►The inequalities in (8.10.1) and (8.10.2) are reversed when

a\geq 1

. …For further inequalities of these types see Qi and Mei (1999) and Neuman (2013). … ►Next, define ►

8.10.7

I=\int_{0}^{x}t^{a-1}e^{t}\,\mathrm{d}t=\Gamma\left(a\right)x^{a}\gamma^{*}% \left(a,-x\right),

\Re a>0

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Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\gamma^{*}\left(\NVar{a},\NVar{z}\right)$ : incomplete gamma function, $\int$ : integral, $\Re$ : real part, $x$ : real variable, $a$ : parameter and $I$ : integral
Permalink:: http://dlmf.nist.gov/8.10.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §8.10, §8.10 and Ch.8

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18: Bibliography N

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W. J. Nellis and B. C. Carlson (1966) Reduction and evaluation of elliptic integrals. Math. Comp. 20 (94), pp. 223–231.

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External Links:: FullText, MathReview (I. A. Stegun), ZentralBlatt
Referenced by:: §19.37(iv).
Encodings:: BibTeX

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E. Neuman (1969a) Elliptic integrals of the second and third kinds. Zastos. Mat. 11, pp. 99–102.

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Notes:: Includes Algol program.
External Links:: MathReview
Referenced by:: §19.39(iii).
Encodings:: BibTeX

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E. Neuman (2004) Inequalities involving Bessel functions of the first kind. JIPAM. J. Inequal. Pure Appl. Math. 5 (4), pp. Article 94, 4 pp. (electronic).

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External Links:: ISSN 1443-5756, FullText, MathReview, ZentralBlatt
Referenced by:: §10.14.
Encodings:: BibTeX

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E. Neuman (2013) Inequalities and bounds for the incomplete gamma function. Results Math. 63 (3-4), pp. 1209–1214.

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External Links:: ISSN 1422-6383, Document, FullText, MathReview (Pierpaolo Natalini), ZentralBlatt
Referenced by:: §8.10, §8.10.
Encodings:: BibTeX

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G. Nikolov and V. Pillwein (2015) An extension of Turán’s inequality. Math. Inequal. Appl. 18 (1), pp. 321–335.

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External Links:: ISSN 1331-4343, Document, Link, MathReview (Gradimir V. Milovanovic), ZentralBlatt
Referenced by:: §18.14(ii).
Encodings:: BibTeX

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… ►This symmetry led to the development of symmetric elliptic integrals, which are free from the transformations of modulus and amplitude that complicate the Legendre theory. Symmetric integrals and their degenerate cases allow greatly shortened integral tables and improved algorithms for numerical computation. Also, the homogeneity of the

R

-function has led to a new type of mean value for several variables, accompanied by various inequalities. … ►This invariance usually replaces sets of twelve equations by sets of three equations and applies also to the relation between the first symmetric elliptic integral and the Jacobian functions. … ►

19 Elliptic Integrals

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20: Edward Neuman

… ►Neuman has published several papers on approximations and expansions, special functions, and mathematical inequalities. …