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Cauchy–Schwarz inequalities for sums and integrals

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1: 1.7 Inequalities
CauchySchwarz Inequality
Minkowski’s Inequality
CauchySchwarz Inequality
Minkowski’s Inequality
§1.7(iv) Jensen’s Inequality
2: Bibliography G
  • L. Gårding (1947) The solution of Cauchy’s problem for two totally hyperbolic linear differential equations by means of Riesz integrals. Ann. of Math. (2) 48 (4), pp. 785–826.
  • R. E. Gaunt (2014) Inequalities for modified Bessel functions and their integrals. J. Math. Anal. Appl. 420 (1), pp. 373–386.
  • A. Gervois and H. Navelet (1984) Some integrals involving three Bessel functions when their arguments satisfy the triangle inequalities. J. Math. Phys. 25 (11), pp. 3350–3356.
  • 3: 4.10 Integrals
    4.10.7 0 x d t ln t = li ( x ) , x > 1 .
    4: 1.9 Calculus of a Complex Variable
    Cauchy’s Integral Formula
    1.9.68 C n = 0 f n ( z ) d z = n = 0 C f n ( z ) d z
    1.9.69 a b n = 0 | f n ( t ) | d t < ,
    1.9.70 n = 0 a b | f n ( t ) | d t < .
    1.9.71 a b n = 0 f n ( t ) d t = n = 0 a b f n ( t ) d t .
    5: 1.4 Calculus of One Variable
    Definite integrals over the Stieltjes measure d α ( x ) could represent a sum, an integral, or a combination of the two. …
    1.4.23_3 a b f ( x ) d α ( x ) = a b w ( x ) f ( x ) d x + n = 1 N w n f ( x n ) .
    Cauchy Principal Values
    1.4.24 a b f ( x ) d x = 𝑃 a b f ( x ) d x = lim ϵ 0 + ( a c ϵ f ( x ) d x + c + ϵ b f ( x ) d x ) ,
    1.4.25 f ( x ) d x = 𝑃 f ( x ) d x = lim b b b f ( x ) d x ,
    6: 6.2 Definitions and Interrelations
    6.2.5 Ei ( x ) = x e t t d t = x e t t d t ,
    6.2.8 li ( x ) = 0 x d t ln t = Ei ( ln x ) , x > 1 .
    7: 19.21 Connection Formulas
    19.21.11 6 R G ( x , y , z ) = 3 ( x + y + z ) R F ( x , y , z ) x 2 R D ( y , z , x ) = x ( y + z ) R D ( y , z , x ) ,
    8: 19.6 Special Cases
    Circular and hyperbolic cases, including Cauchy principal values, are unified by using R C ( x , y ) . … For the Cauchy principal value of Π ( ϕ , α 2 , k ) when α 2 > c , see §19.7(iii). …
    9: 19.20 Special Cases
    If x , y , z are permuted so that 0 x < y < z , then the Cauchy principal value of R J is given by …
    10: 19.16 Definitions
    §19.16(i) Symmetric Integrals
    In (19.16.2) the Cauchy principal value is taken when p is real and negative. … All other elliptic cases are integrals of the second kind. … Each of the four complete integrals (19.16.20)–(19.16.23) can be integrated to recover the incomplete integral: …