Jensen inequality for integrals

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21: Bibliography

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Monotonicity theorems and inequalities for the complete elliptic integrals. J. Comput. Appl. Math. 172 (2), pp. 289–312.

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External Links:

ISSN 0377-0427, Document, MathReview (Živorad Tomovski), ZentralBlatt

Referenced by:

§19.9(i).

Encodings:

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On some inequalities for the incomplete gamma function. Math. Comp. 66 (218), pp. 771–778.

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External Links:

ISSN 0025-5718, Document, MathReview, ZentralBlatt

Referenced by:

§8.10.

Encodings:

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Functional inequalities for complete elliptic integrals and their ratios. SIAM J. Math. Anal. 21 (2), pp. 536–549.

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ISSN 0036-1410, Document, MathReview (K. C. Gupta), ZentralBlatt

Referenced by:

§19.9(i).

Encodings:

BibTeX

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Functional inequalities for hypergeometric functions and complete elliptic integrals. SIAM J. Math. Anal. 23 (2), pp. 512–524.

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ISSN 0036-1410, Document, MathReview (J. M. H. Peters), ZentralBlatt

Referenced by:

§19.9(i).

Encodings:

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Inequalities for elliptic integrals. Publ. Inst. Math. (Beograd) (N.S.) 37(51), pp. 61–63.

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External Links:

ISSN 0350-1302, MathReview, ZentralBlatt

Referenced by:

§19.9(i).

Encodings:

BibTeX

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22: 4.32 Inequalities

§4.32 Inequalities

… ►For these and other inequalities involving hyperbolic functions see Mitrinović (1964, pp. 61, 76, 159) and Mitrinović (1970, p. 270).

23: Bibliography G

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An inequality of Turán type for Jacobi polynomials. Proc. Amer. Math. Soc. 32, pp. 435–439.

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External Links:

ISSN 0002-9939, Document, MathReview (A. E. Danese), ZentralBlatt

Referenced by:

§18.14(ii).

Encodings:

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New inequalities for the zeros of Jacobi polynomials. SIAM J. Math. Anal. 18 (6), pp. 1549–1562.

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External Links:

ISSN 0036-1410, Document, MathReview (E. Riekstiņš), ZentralBlatt

Referenced by:

§18.16(ii), §18.16(vi).

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Inequalities for modified Bessel functions and their integrals. J. Math. Anal. Appl. 420 (1), pp. 373–386.

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ISSN 0022-247X, Document, Link, MathReview (Nihat Yagmur), ZentralBlatt

Referenced by:

§10.37, §10.37.

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A harmonic mean inequality for the gamma function. SIAM J. Math. Anal. 5 (2), pp. 278–281.

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Document, MathReview (S. Saran), ZentralBlatt

Referenced by:

§5.6(i).

Encodings:

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Some integrals involving three Bessel functions when their arguments satisfy the triangle inequalities. J. Math. Phys. 25 (11), pp. 3350–3356.

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ISSN 0022-2488, Document, MathReview (B. R. Bhonsle), ZentralBlatt

Referenced by:

§10.22(iv), §10.43(iv).

Encodings:

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24: Bibliography L

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Inequalities and approximations for zeros of Bessel functions of small order. SIAM J. Math. Anal. 14 (2), pp. 383–388.

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External Links:

ISSN 0036-1410, Document, MathReview (S. Ahmed), ZentralBlatt

Referenced by:

§10.21(iv).

Encodings:

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Monotonicity results and inequalities for the gamma and error functions. J. Comput. Appl. Math. 23 (1), pp. 25–33.

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ISSN 0377-0427, Document, MathReview (M. E. Cohen), ZentralBlatt

Referenced by:

§7.8.

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Further inequalities for the gamma function. Math. Comp. 42 (166), pp. 597–600.

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External Links:

ISSN 0025-5718, FullText, MathReview (H. E. Fettis), ZentralBlatt

Referenced by:

§5.6(i).

Encodings:

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Inequalities for Bessel functions. J. Comput. Appl. Math. 15 (1), pp. 75–81.

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ISSN 0377-0427, Document, MathReview (B. D. Agrawal), ZentralBlatt

Referenced by:

§10.14.

Encodings:

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Inequalities for ultraspherical polynomials and the gamma function. J. Approx. Theory 40 (2), pp. 115–120.

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External Links:

ISSN 0021-9045, Document, MathReview (B. R. Bhonsle), ZentralBlatt

Referenced by:

§18.14(i).

Encodings:

BibTeX

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25: 1.8 Fourier Series

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1.8.5 $$\frac{1}{\pi }{\int }_{-\pi }^{\pi }{\left|f(x)\right|}^{2}dx=\frac{1}{2}{\left|a_0\right|}^2+\sum _{n=1}^{\mathrm{\infty }}({\left|a_n\right|}^2+{\left|b_n\right|}^2),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\int$: integral, $n$: nonnegative integer and $\left|x\right|$: absolute value of $x$

Referenced by:

Erratum (V1.2.0) for Equations (1.8.5), (1.8.6)

Permalink:

http://dlmf.nist.gov/1.8.E5

Encodings:

pMML, png, TeX

Modification (effective with 1.2.0):

Previously this equation was given as an inequality. For square integrable functions this inequality $\ge$ can be sharpened to $=$.

See also:

Annotations for §1.8(i), §1.8(i), §1.8 and Ch.1

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1.8.6 $$\frac{1}{2\pi }{\int }_{-\pi }^{\pi }{\left|f(x)\right|}^{2}dx=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{\left|c_n\right|}^2,$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\int$: integral, $n$: nonnegative integer and $\left|x\right|$: absolute value of $x$

Referenced by:

§1.8(i), Erratum (V1.2.0) for Equations (1.8.5), (1.8.6)

Permalink:

http://dlmf.nist.gov/1.8.E6

Encodings:

pMML, png, TeX

Modification (effective with 1.2.0):

Previously this equation was given as an inequality. For square integrable functions this inequality $\ge$ can be sharpened to $=$.

See also:

Annotations for §1.8(i), §1.8(i), §1.8 and Ch.1

… ►(1.8.10) continues to apply if either $a$ or $b$ or both are infinite and/or $f(x)$ has finitely many singularities in $(a,b)$, provided that the integral converges uniformly (§1.5(iv)) at $a,b$, and the singularities for all sufficiently large $\lambda$. … ►It follows from definition (1.14.1) that the integral in (1.8.14) is equal to $\sqrt{2\pi }ℱ\left(f\right)\left(-2\pi n\right)$. … ►

1.8.15 $$\frac{1}{2}f(0)+\sum _{n=1}^{\mathrm{\infty }}f(n)={\int }_0^{\mathrm{\infty }}f(x)dx+2\sum _{n=1}^{\mathrm{\infty }}{\int }_0^{\mathrm{\infty }}f(x)\cos \left(2\pi nx\right)dx.$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\int$: integral and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/1.8.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §1.8(iv), §1.8 and Ch.1

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26: Bibliography P

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An inequality for the Bessel function $J_{\nu }(\nu x)$ . SIAM J. Math. Anal. 15 (1), pp. 203–205.

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External Links:

ISSN 0036-1410, Document, MathReview (L. Debnath)

Referenced by:

§10.14, §10.37.

Encodings:

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Numerical calculation of the generalized Fermi-Dirac integrals. Comput. Phys. Comm. 55 (2), pp. 127–136.

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External Links:

Document

Referenced by:

§25.18(i).

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Automatic computation of Bessel function integrals. Comput. Phys. Comm. 25 (3), pp. 289–295.

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Notes:

Double-precision Fortran, maximum accuracy 20S.

External Links:

ISSN 0010-4655, Document, Code

Referenced by:

6th item.

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Inequalities for the zeros of the Airy functions. SIAM J. Math. Anal. 22 (1), pp. 260–267.

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External Links:

ISSN 0036-1410, Document, MathReview, ZentralBlatt

Referenced by:

§9.9(iv).

Encodings:

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Elliptic integrals. Computers in Physics 4 (1), pp. 92–96.

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Referenced by:

2nd item.

Encodings:

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27: 19.14 Reduction of General Elliptic Integrals

§19.14 Reduction of General Elliptic Integrals

… ►(These four cases include 12 integrals in Abramowitz and Stegun (1964, p. 596).) … ►Legendre (1825–1832) showed that every elliptic integral can be expressed in terms of the three integrals in (19.1.2) supplemented by algebraic, logarithmic, and trigonometric functions. …The choice among 21 transformations for final reduction to Legendre’s normal form depends on inequalities involving the limits of integration and the zeros of the cubic or quartic polynomial. …

28: 10.37 Inequalities; Monotonicity

§10.37 Inequalities; Monotonicity

… ►For sharper inequalities when the variables are real see Paris (1984) and Laforgia (1991). …

29: 18.14 Inequalities

§18.14 Inequalities

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Legendre

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Jacobi

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Szegő–Szász Inequality

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30: 19.30 Lengths of Plane Curves

§19.30 Lengths of Plane Curves

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§19.30(i) Ellipse

… ►Approximations and inequalities for $L(a,b)$ are given in §19.9(i). … ►

§19.30(ii) Hyperbola

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