In mathematics, a function f is logarithmically convex or superconvex if log ∘ f {\displaystyle {\log }\circ f} , the composition of the logarithm with...

Karamata's inequality Logarithmically convex function Pseudoconvex function Quasiconvex function Subderivative of a convex function "Lecture Notes 2" (PDF)...

In convex analysis, a non-negative function f : Rn → R+ is logarithmically concave (or log-concave for short) if its domain is a convex set, and if it...

concave function is one for which the function value at any convex combination of elements in the domain is greater than or equal to that convex combination...

positive and logarithmically convex for positive z and whose value at 1 is 1 (a function is logarithmically convex if its logarithm is convex). Another characterisation...

one in terms of logarithmically convex functions, one in terms of increasing functions, and one in terms of non-negative functions. The conjecture has...

condition is required, which is called logarithmically-convex. A Reinhardt domain D is called logarithmically convex if the image λ ( D ∗ ) {\displaystyle...

of a quasiconvex function that is neither convex nor continuous. Convex function Concave function Logarithmically concave function Pseudoconvexity in...

measures is log-concave. Convex measure, a generalisation of this concept Logarithmically concave function Prékopa, A. (1980). "Logarithmic concave measures and...

writing in 1896; Hadamard published no proof. Maximum principle Logarithmically convex function Hardy's theorem Hadamard three-line theorem Borel–Carathéodory...

of barrier functions are inverse barrier functions and logarithmic barrier functions. Resumption of interest in logarithmic barrier functions was motivated...

} Thus, a measure being 0-convex is the same thing as it being a logarithmically concave measure. The classes of s-convex measures form a nested increasing...

be logarithmically convex, if M k + 1 / M k {\displaystyle M_{k+1}/M_{k}} is increasing. When M k {\displaystyle M_{k}} is logarithmically convex, then...

power series is not necessarily an open ball; these regions are logarithmically-convex Reinhardt domains, the simplest example of which is a polydisk....

_{i}(\mathbf {x} ,\mathbf {y} ))} is the partition function. This density is a logarithmically convex function, and thus the common inference task in PSL of...

In geometry, the convex hull, convex envelope or convex closure of a shape is the smallest convex set that contains it. The convex hull may be defined...

asymptotes as x → ± ∞ {\displaystyle x\rightarrow \pm \infty } . A sigmoid function is convex for values less than a particular point, and it is concave for values...

which shows that boosting performs gradient descent in a function space using a convex cost function. Given images containing various known objects in the...

completely monotonic function, logarithmically completely monotonic function, strongly logarithmically completely monotonic function, strongly completely...

this formula internally. LSE is convex but not strictly convex. We can define a strictly convex log-sum-exp type function by adding an extra argument set...

λ i {\displaystyle \lambda _{i}} are all of the eigenvalues. Is a convex function of ρ {\displaystyle \rho } : N ( ∑ i p i ρ i ) ≤ ∑ i p i N ( ρ i )...

function of the shape parameters α and β. § Moments of logarithmically transformed random variables contains formulas for moments of logarithmically transformed...

In mathematics, logarithmic Sobolev inequalities are a class of inequalities involving the norm of a function f, its logarithm, and its gradient ∇ f {\displaystyle...

self-concordant barrier is a particular self-concordant function, that is also a barrier function for a particular convex set. Self-concordant barriers are important...

the term bidisc is sometimes used. A polydisc is an example of logarithmically convex Reinhardt domain. Poincare, H, Les fonctions analytiques de deux...

functions that satisfy these conditions, but the gamma function is the unique one that is meromorphic in the whole complex plane, and logarithmically...

two-dimensional graph of numerical data that uses logarithmic scales on both the horizontal and vertical axes. Power functions – relationships of the form y = a x k...

x f (x) for x > 0 and  f  is logarithmically convex. A treatment of this theorem is in Artin's book The Gamma Function, which has been reprinted by the...

a convex function and G is a convex set. Without loss of generality, we can assume that the objective f is a linear function. Usually, the convex set...

motor driven by direct current. Decibel A logarithmic unit of ratios. Definite integral The integral of a function between an upper and lower limit. Deflection...