inequalities

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21: Bibliography N

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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.

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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:

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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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22: 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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A harmonic mean inequality for the gamma function. J. Comput. Appl. Math. 87 (2), pp. 195–198.

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Notes:

For corrigendum see same journal v. 90 (1998), no. 2, p. 265

External Links:

ISSN 0377-0427, Document, MathReview, ZentralBlatt

Referenced by:

§5.6(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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Gamma function inequalities. Numer. Algorithms 49 (1-4), pp. 53–84.

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External Links:

ISSN 1017-1398, Document, FullText, MathReview (Junesang Choi), ZentralBlatt

Referenced by:

§5.6(i), §5.6(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:

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23: Bille C. Carlson

… ►Also, the homogeneity of the $R$-function has led to a new type of mean value for several variables, accompanied by various inequalities. …

24: 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:

BibTeX

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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).

Encodings:

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New inequalities for the zeros of confluent hypergeometric functions. In Asymptotic and computational analysis (Winnipeg, MB, 1989), pp. 175–192.

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External Links:

MathReview, ZentralBlatt

Referenced by:

§13.9(i), §13.9(ii).

Encodings:

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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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External Links:

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

Referenced by:

§10.37, §10.37.

Encodings:

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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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External Links:

Document, MathReview (S. Saran), ZentralBlatt

Referenced by:

§5.6(i).

Encodings:

BibTeX

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25: 1.3 Determinants, Linear Operators, and Spectral Expansions

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Hadamard’s Inequality

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26: 18.16 Zeros

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Inequalities

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Inequalities

… ►The constant $j_{\alpha ,m}^2$ in (18.16.10) is the best possible since the ratio of the two sides of this inequality tends to 1 as $n\to \mathrm{\infty }$. … ►when $\alpha \notin (-\frac{1}{2},\frac{1}{2})$. …

27: 18.38 Mathematical Applications

… ►The Askey–Gasper inequality …also the case $\beta =0$ of (18.14.26), was used in de Branges’ proof of the long-standing Bieberbach conjecture concerning univalent functions on the unit disk in the complex plane. …

28: 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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External Links:

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

Referenced by:

§7.8.

Encodings:

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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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External Links:

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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29: 13.9 Zeros

… ►Inequalities for ${\varphi }_r$ are given in Gatteschi (1990), and identities involving infinite series of all of the complex zeros of $M(a,b,x)$ are given in Ahmed and Muldoon (1980). … ►Inequalities for the zeros of $U(a,b,x)$ are given in Gatteschi (1990). …

30: 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

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