Hadamard inequality

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21: 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): …

22: 25.10 Zeros

… ►Also, $\zeta \left(s\right)\ne 0$ for $\mathrm{\Re }s=1$, a property first established in Hadamard (1896) and de la Vallée Poussin (1896a, b) in the proof of the prime number theorem (25.16.3). …

23: 8.10 Inequalities

§8.10 Inequalities

… ►The inequalities in (8.10.1) and (8.10.2) are reversed when $a\ge 1$. …For further inequalities of these types see Qi and Mei (1999) and Neuman (2013). …

24: 16.23 Mathematical Applications

… ►In the proof of this conjecture de Branges (1985) uses the inequality …The proof of this inequality is given in Askey and Gasper (1976). …

25: 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

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§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.

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An extension of Turán’s inequality. Math. Inequal. Appl. 18 (1), pp. 321–335.

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ISSN 1331-4343, Document, Link, MathReview (Gradimir V. Milovanovic), ZentralBlatt

Referenced by:

§18.14(ii).

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26: 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).

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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).

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

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ISSN 0025-5718, Document, MathReview, ZentralBlatt

Referenced by:

§8.10.

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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).

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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).

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27: 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. …

28: 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).

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

Referenced by:

§18.16(ii), §18.16(vi).

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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).

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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.

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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).

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29: 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 $\alpha \notin (-\frac{1}{2},\frac{1}{2})$. …

30: 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. …