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1: 10.37 Inequalities; Monotonicity
§10.37 Inequalities; Monotonicity
2: 10.14 Inequalities; Monotonicity
§10.14 Inequalities; Monotonicity
For monotonicity properties of J ν ( ν ) and J ν ( ν ) see Lorch (1992). … For further monotonicity properties see Landau (1999, 2000), and Muldoon and Spigler (1984).
3: 35.9 Applications
For other statistical applications of F q p functions of matrix argument see Perlman and Olkin (1980), Groeneboom and Truax (2000), Bhaumik and Sarkar (2002), Richards (2004) (monotonicity of power functions of multivariate statistical test criteria), Bingham et al. (1992) (Procrustes analysis), and Phillips (1986) (exact distributions of statistical test criteria). …
4: Bibliography L
  • A. Laforgia and M. E. Muldoon (1988) Monotonicity properties of zeros of generalized Airy functions. Z. Angew. Math. Phys. 39 (2), pp. 267–271.
  • A. Laforgia and S. Sismondi (1988) Monotonicity results and inequalities for the gamma and error functions. J. Comput. Appl. Math. 23 (1), pp. 25–33.
  • L. J. Landau (2000) Bessel functions: Monotonicity and bounds. J. London Math. Soc. (2) 61 (1), pp. 197–215.
  • J. T. Lewis and M. E. Muldoon (1977) Monotonicity and convexity properties of zeros of Bessel functions. SIAM J. Math. Anal. 8 (1), pp. 171–178.
  • L. Lorch, M. E. Muldoon, and P. Szegő (1970) Higher monotonicity properties of certain Sturm-Liouville functions. III. Canad. J. Math. 22, pp. 1238–1265.
  • 5: 1.4 Calculus of One Variable
    §1.4(i) Monotonicity
    Each of the preceding four cases is classified as monotonic; sometimes strictly monotonic is used for the strictly increasing or strictly decreasing cases. … For f ( x ) monotonic and ϕ ( x ) integrable on [ a , b ] , there exists c [ a , b ] , such that …
    6: 8.13 Zeros
    The negative zero x - ( a ) decreases monotonically in the interval - 1 < a < 0 , and satisfies …
    7: 9.8 Modulus and Phase
    §9.8(iii) Monotonicity
    8: 8.3 Graphics
    Some monotonicity properties of γ * ( a , x ) and Γ ( a , x ) in the four quadrants of the ( a , x )-plane in Figure 8.3.6 are given in Erdélyi et al. (1953b, §9.6). …
    9: 23.20 Mathematical Applications
    The boundary of the rectangle R , with vertices 0 , ω 1 , ω 1 + ω 3 , ω 3 , is mapped strictly monotonically by onto the real line with 0 , ω 1 e 1 , ω 1 + ω 3 e 2 , ω 3 e 3 , 0 - . … The two pairs of edges [ 0 , ω 1 ] [ ω 1 , 2 ω 3 ] and [ 2 ω 3 , 2 ω 3 - ω 1 ] [ 2 ω 3 - ω 1 , 0 ] of R are each mapped strictly monotonically by onto the real line, with 0 , ω 1 e 1 , 2 ω 3 - ; similarly for the other pair of edges. …
    10: 3.8 Nonlinear Equations
  • (a)

    f ( x 0 ) f ′′ ( x 0 ) > 0 and f ( x ) , f ′′ ( x ) do not change sign between x 0 and ξ (monotonic convergence).

  • (b)

    f ( x 0 ) f ′′ ( x 0 ) < 0 , f ( x ) , f ′′ ( x ) do not change sign in the interval ( x 0 , x 1 ) , and ξ [ x 0 , x 1 ] (monotonic convergence after the first iteration).