Cauchy?Schwarz inequalities for sums and integrals

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1: 1.7 Inequalities

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Cauchy–Schwarz Inequality

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Minkowski’s Inequality

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Cauchy–Schwarz Inequality

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Minkowski’s Inequality

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§1.7(iv) Jensen’s Inequality

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2: 6.8 Inequalities

§6.8 Inequalities

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3: 19.24 Inequalities

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§19.24(i) Complete Integrals

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§19.24(ii) Incomplete Integrals

… ►Inequalities for $R_C(x,y)$ and $R_D(x,y,z)$ are included as special cases (see (19.16.6) and (19.16.5)). ►Other inequalities for $R_F(x,y,z)$ are given in Carlson (1970). …

4: 8.19 Generalized Exponential Integral

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8.19.7 $$E_n\left(z\right)=\frac{{(-z)}^{n-1}}{(n-1)!}{E}_1\left(z\right)+\frac{{\mathrm{e}}^{-z}}{(n-1)!}\sum _{k=0}^{n-2}(n-k-2)!{(-z)}^k,$$ $n=2,3,\mathrm{\dots }$.

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Symbols:

$\mathrm{e}$: base of natural logarithm, $E_1\left(z\right)$: exponential integral, $E_p\left(z\right)$: generalized exponential integral, $!$: factorial (as in $n!$), $z$: complex variable, $k$: nonnegative integer and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/8.19.E7

Encodings:

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See also:

Annotations for §8.19(iii), §8.19 and Ch.8

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8.19.8 $$E_n\left(z\right)=\frac{{(-z)}^{n-1}}{(n-1)!}(\psi \left(n\right)-\ln z)-\sum _{\begin{array}{c}k=0\\ k\ne n-1\end{array}}^{\mathrm{\infty }}\frac{{(-z)}^k}{k!(1-n+k)},$$

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Symbols:

$\psi \left(z\right)$: psi (or digamma) function, $E_p\left(z\right)$: generalized exponential integral, $!$: factorial (as in $n!$), $\ln z$: principal branch of logarithm function, $z$: complex variable, $k$: nonnegative integer and $n$: nonnegative integer

A&S Ref:

5.1.12

Referenced by:

§8.19(iv)

Permalink:

http://dlmf.nist.gov/8.19.E8

Encodings:

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See also:

Annotations for §8.19(iv), §8.19 and Ch.8

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8.19.10 $$E_p\left(z\right)=z^{p-1}\mathrm{\Gamma }\left(1-p\right)-\sum _{k=0}^{\mathrm{\infty }}\frac{{(-z)}^k}{k!(1-p+k)},$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $E_p\left(z\right)$: generalized exponential integral, $!$: factorial (as in $n!$), $z$: complex variable, $p$: parameter and $k$: nonnegative integer

Referenced by:

§8.19(iv)

Permalink:

http://dlmf.nist.gov/8.19.E10

Encodings:

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See also:

Annotations for §8.19(iv), §8.19 and Ch.8

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§8.19(ix) Inequalities

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8.19.24 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-at}{E}_n\left(t\right)dt=\frac{{(-1)}^{n-1}}{a^n}\left(\ln \left(1+a\right)+\sum _{k=1}^{n-1}\frac{{(-1)}^k{a}^k}{k}\right),$$ $n=1,2,\mathrm{\dots }$, $\mathrm{\Re }a>-1$,

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Symbols:

$dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $E_p\left(z\right)$: generalized exponential integral, $\int$: integral, $\ln z$: principal branch of logarithm function, $\mathrm{\Re }$: real part, $a$: parameter, $k$: nonnegative integer and $n$: nonnegative integer

A&S Ref:

5.1.34

Referenced by:

§8.19(x)

Permalink:

http://dlmf.nist.gov/8.19.E24

Encodings:

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See also:

Annotations for §8.19(x), §8.19 and Ch.8

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5: Bibliography Q

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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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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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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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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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6: 7.8 Inequalities

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7.8.7 $x>0$.

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Symbols:

$F\left(z\right)$: Dawson’s integral, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\sinh z$: 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:

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Addition (effective with 1.1.5):

A new inequality was added.

See also:

Annotations for §7.8 and Ch.7

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7: 19.21 Connection Formulas

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19.21.11 $$6R_G(x,y,z)=3(x+y+z)R_F(x,y,z)-\sum x^2{R}_D(y,z,x)=\sum x(y+z)R_D(y,z,x),$$

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Symbols:

$R_D(x,y,z)$: elliptic integral symmetric in only two variables, $R_F(x,y,z)$: symmetric elliptic integral of first kind and $R_G(x,y,z)$: symmetric elliptic integral of second kind

Referenced by:

§19.21(i), §19.21(ii)

Permalink:

http://dlmf.nist.gov/19.21.E11

Encodings:

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See also:

Annotations for §19.21(ii), §19.21 and Ch.19

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8: 16.20 Integrals and Series

§16.20 Integrals and Series

►Integrals of the Meijer $G$-function are given in Apelblat (1983, §19), Erdélyi et al. (1953a, §5.5.2), Erdélyi et al. (1954a, §§6.9 and 7.5), Luke (1969a, §3.6), Luke (1975, §5.6), Mathai (1993, §3.10), and Prudnikov et al. (1990, §2.24). … ►

9: 19.9 Inequalities

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§19.9(i) Complete Integrals

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§19.9(ii) Incomplete Integrals

… ►Simple inequalities for incomplete integrals follow directly from the defining integrals (§19.2(ii)) together with (19.6.12): … ►Sharper inequalities for $F(\varphi ,k)$ are: … ►Other inequalities for $F(\varphi ,k)$ can be obtained from inequalities for $R_F(x,y,z)$ given in Carlson (1966, (2.15)) and Carlson (1970) via (19.25.5).

10: 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:

BibTeX

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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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External Links:

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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External Links:

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

Referenced by:

§19.9(i).

Encodings:

BibTeX

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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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Jacobi polynomials. I. New proofs of Koornwinder’s Laplace type integral representation and Bateman’s bilinear sum. SIAM J. Math. Anal. 5, pp. 119–124.

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External Links:

ISSN 1095-7154, Document, MathReview (R. P. Soni), ZentralBlatt

Referenced by:

§18.18(iv).

Encodings:

BibTeX

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