{"id":22447,"date":"2026-03-03T03:03:00","date_gmt":"2026-03-03T02:03:00","guid":{"rendered":"https:\/\/djalil.chafai.net\/blog\/?p=22447"},"modified":"2026-02-09T10:40:24","modified_gmt":"2026-02-09T09:40:24","slug":"convexity-of-entropies-and-integral-representation","status":"publish","type":"post","link":"https:\/\/djalil.chafai.net\/blog\/2026\/03\/03\/convexity-of-entropies-and-integral-representation\/","title":{"rendered":"Convexity of entropies and integral representation"},"content":{"rendered":"<figure id=\"attachment_22448\" aria-describedby=\"caption-attachment-22448\" style=\"width: 250px\" class=\"wp-caption aligncenter\"><a href=\"https:\/\/en.wikipedia.org\/wiki\/Constantin_Carath%C3%A9odory\"><img loading=\"lazy\" src=\"http:\/\/djalil.chafai.net\/blog\/wp-content\/uploads\/2026\/01\/caratheodory.jpg\" alt=\"Photo of Constantin Carath\u00e9odory\" width=\"250\" height=\"300\" class=\"size-full wp-image-22448\" \/><\/a><figcaption id=\"caption-attachment-22448\" class=\"wp-caption-text\">Constantin Carath\u00e9odory (1873-1950), German mathematician of Greek origin. Started as an Engineer. He contributed to many fields including measure theory, complex analysis, and convex geometry<\/figcaption><\/figure>\n<p style=\"text-align:justify;\">This post is about the structure of the set of $\\Phi$ for which the $\\Phi$-entropy is convex. It is a follow-up of a <a href=\"https:\/\/djalil.chafai.net\/blog\/2025\/01\/25\/about-variance-and-entropy\/\">previous post<\/a> published one year ago about variance and entropy. We connect it to a theorem by Stieltjes and Nevanlinna on an integral representation of certain smooth functions, which is the subject of <a href=\"https:\/\/djalil.chafai.net\/blog\/2026\/01\/23\/herglotz-and-nevanlinna-integral-representations\/\">another previous post<\/a>.<\/p>\n<p style=\"text-align:justify;\"><strong>$\\Phi$-entropy and convexity.<\/strong> Let us consider a convex and $\\mathcal{C}^4$ function $\\Phi:I\\to\\mathbb{R}$ on an open interval $I\\subset\\mathbb{R}$. For every probability space $(\\Omega,\\mathcal{A},\\mu)$, the following set is convex: \\[   L^\\Phi(\\mu)=\\{f:\\Omega\\to I\\mid f\\in L^1(\\mu),\\Phi(f)\\in L^1(\\mu)\\} \\] The $\\Phi$-entropy of $f\\in L^\\Phi(\\mu)$ is defined by \\[   \\mathrm{Ent}_\\mu^\\Phi(f)=\\int\\Phi(f)\\mathrm{d}\\mu-\\Phi\\Bigl(\\int f\\mathrm{d}\\mu\\Bigr). \\] Moreover the following conditions are equivalent:<\/p>\n<p style=\"text-align:justify;\">\n<ul>\n<li>$f\\mapsto\\mathrm{Ent}_\\mu^\\Phi(f)$ is convex for all $(\\Omega,\\mathcal{A},\\mu)$. <\/li>\n<li>$(u,v)\\mapsto\\Phi''(u)v^2$ is convex <\/li>\n<li>either $\\Phi$ is affine, or $\\Phi'' > 0$ and $1\/\\Phi''$ is concave (in   other words $\\Phi''\\Phi''''\\geq 2\\Phi'''^2$). <\/li>\n<\/ul>\n<p style=\"text-align:justify;\">We denote by $\\mathcal{K}$ the set of such $\\Phi$. For simplicity, we take from now on $I=\\mathbb{R}_+=(0,+\\infty)$.<\/p>\n<p style=\"text-align:justify;\"><strong>Examples.<\/strong> The set $\\mathcal{K}$ is convex, and contains all the affine functions. Basic non-affine examples of elements of $\\mathcal{K}$ are<\/p>\n<p style=\"text-align:justify;\">\n<ul>\n<li>$\\Phi(u)=u\\log(u)$ <\/li>\n<li>$\\Phi(u)=u^p$, $1 < p\\leq 2$. <\/li>\n<\/ul>\n<p style=\"text-align:justify;\">The set $\\mathcal{K}$ is a convex cone in the sense that it is stable by linear combinations with non-negative coefficients. Since it contains all affine functions, we have $u\\mapsto\\frac{u^p-u}{p-1}\\in\\mathcal{K}$ for all $1 < p\\leq 2$, and we recover $u\\mapsto u\\log(u)$ as $p\\to1$.<\/p>\n<p style=\"text-align:justify;\">The case $u\\mapsto u^p$, $1 < p&lt;2$ is not extreme in the sense that it is a mixture (convex conic combination) of affine functions and shifts of $u\\mapsto u\\log(u)$, more precisely \\[   u^p=c_p\\int_0^\\infty t^{p-2}\\varphi_t(u)\\mathrm{d}t \\] where \\begin{align*}   c_p&=\\frac{p(p-1)\\sin(\\pi(p-1))}{\\pi}\\\\   \\varphi_t(u)&=(u+t)\\log(u+t)\\underbrace{-(1+\\log(t))u-t\\log(t)}_{\\text{affine}} \\end{align*} Indeed, this comes from the Stieltjes identity,  \\[   \\int_0^\\infty\\frac{t^{a-1}}{u+t}\\mathrm{d}t=\\frac{\\pi}{\\sin(\\pi a)}u^{a-1},   \\quad 0 < a&lt;1,\\quad u>0, \\] which boils down to an Euler Beta integral via the substitutions $s=t\/u$, $x=1\/(1+s)$ as \\begin{align*}   \\int_0^\\infty\\frac{t^{a-1}}{u+t}\\mathrm{d}t   &=u^{a-1}\\int_0^\\infty\\frac{s^{a-1}}{1+s}\\mathrm{d}s\\\\   &=u^{a-1}\\int_0^1(1-x)^{a-1}x^{-a}\\mathrm{d}x   =u^{a-1}\\frac{\\Gamma(a)\\Gamma(1-a)}{\\Gamma(1)}   =u^{a-1}\\frac{\\pi}{\\sin(\\pi a)}   \\end{align*} where the last equality follows from the Euler reflection formula.<\/p>\n<p style=\"text-align:justify;\">We could ask about the extremality of $u\\mapsto u\\log(u)$ and $u\\mapsto u^2$, and some sort of integral representation of elements of $\\mathcal{K}$ as mixtures of extreme points.<\/p>\n<p style=\"text-align:justify;\"><strong>Extremality.<\/strong> The vector space of affine functions $\\mathcal{A}=\\{u\\mapsto au+b:a,b\\in\\mathbb{R}\\}$ is contained in $\\mathcal{K}$. Thus $\\mathcal{K}$ has no extreme points, indeed for all $f\\in\\mathcal{K}$ and $a\\in\\mathcal{A}$, $f=(f+a)+(-a)$ which is the sum of two elements of $\\mathcal{K}$. A natural way to remove the affine part is to work with the second derivative, namely to consider the convex cone \\begin{align*}   \\mathcal{K}''   &=\\{\\varphi=\\Phi'':\\Phi\\in\\mathcal{K}\\}\\\\   &=\\{0\\}\\cup\\{\\varphi\\in\\mathcal{C}^2(\\mathbb{R}_+):\\varphi > 0,\\varphi\\varphi''-2\\varphi'^2\\geq0\\}. \\end{align*}<\/p>\n<p style=\"text-align:justify;\"><strong>Extremality ODE.<\/strong> For all $\\varphi\\in\\mathcal{K}''\\setminus\\{0\\}$, \\[   \\varphi\\text{ is extreme in }\\mathcal{K}''\\quad\\text{iff}\\quad Q:=\\varphi\\varphi''-2\\varphi'^2=0. \\]   <strong>Proof.<\/strong> Suppose that $Q\\neq0$ : $Q(u_0) > 0$ for some $u_0$. Choose a compact interval $J$ and constants $m,\\beta > 0$ with $u_0\\in J$ and $Q\\geq m$ and $\\varphi\\geq \\beta$ on $J$. Let $\\eta\\in C^2_c(J)$, $\\eta\\neq0$. Then \\[   \\varphi=\\tfrac{1}{2}\\varphi_++\\tfrac{1}{2}\\varphi_-   \\quad\\text{where}\\quad   \\varphi_\\pm:=\\varphi\\pm \\varepsilon\\eta,   \\quad\\text{for all $\\varepsilon > 0$}. \\] If $\\varepsilon < \\frac{\\beta}{2\\|\\eta\\|_\\infty}$, we have $\\varphi_\\pm\\geq \\beta\/2 > 0$ on $J$, while $\\varphi_\\pm=\\varphi$ outside $J$. Moreover \\[   Q_{\\pm}   :=\\varphi_\\pm\\varphi_\\pm''-2\\varphi_\\pm'^2   =Q \\pm \\varepsilon L + \\varepsilon^2R, \\] where $L:=\\varphi\\eta''+\\varphi''\\eta-4\\varphi'\\eta'$ and $R:=\\eta\\eta''-2\\eta'^2$ on $J$. Then, on $J$, \\[   Q_\\pm\\geq m-\\varepsilon L_0-\\varepsilon^2R_0   \\quad\\text{where}\\quad   L_0:=\\|L\\|_{\\infty,J}\\quad\\text{and}\\quad R_0:=\\|R\\|_{\\infty,J}. \\] Thus, for small enough $\\varepsilon > 0$, $Q_\\pm\\geq 0$ on $J$. Outside $J$, we have $\\eta=0$ and $Q_\\pm=Q\\geq 0$. Hence $\\varphi_\\pm\\in\\mathcal{K}''$ for small enough $\\varepsilon > 0$, and therefore $\\frac{1}{2}\\varphi_\\pm\\in\\mathcal{K}''$ for small enough $\\varepsilon > 0$.<\/p>\n<p style=\"text-align:justify;\">Let us show that $\\frac{1}{2}\\varphi_\\pm$ are not colinear for small enough $\\varepsilon > 0$. Suppose $\\varphi_+=c\\varphi$ for some $c\\geq 0$. Outside $J$ we have $\\varphi_+=\\varphi$, hence $\\varphi=c\\varphi$ outside $J$. Since $\\varphi > 0$, this forces $c=1$, hence $\\varphi_+=\\varphi$ everywhere and $\\varepsilon\\eta\\equiv 0$, contradiction. Thus $\\varphi_+$ and $\\varphi$ (and thus $\\varphi_-$) are not colinear, and the decomposition $\\varphi=\\frac{1}{2}\\varphi_-+\\frac{1}{2}\\varphi_+$ is not trivial.<\/p>\n<p style=\"text-align:justify;\">Hence $Q\\neq0$ implies that $\\varphi$ is not extreme (equivalently, if $\\varphi$ is extreme then $Q=0$).<\/p>\n<p style=\"text-align:justify;\">Conversely, suppose that $Q=0$, and $\\varphi=\\varphi_1+\\varphi_2$ with $\\varphi_1,\\varphi_2\\in\\mathcal{K}''$. Let us define the bivariate functions $F(u,v):=\\varphi(u)v^2$ and $F_i(u,v):=\\varphi_i(u)v^2$. Then \\[ \\nabla^2F=\\nabla^2F_1+\\nabla^2F_2,\\quad \\nabla^2F_i\\succeq 0. \\] Since $Q=0$, for all $v\\neq 0$, the Hessian matrix \\[   \\nabla^2F(u,v)=\\begin{pmatrix}\\varphi''(u)v^2&2\\varphi'(u)v\\\\2\\varphi'(u)v&2\\varphi(u)\\end{pmatrix} \\] has determinant $0$ and $(2,2)$ entry $2\\varphi(u) > 0$, hence has rank $1$.<\/p>\n<p style=\"text-align:justify;\">Fix $(u,v)$ with $v\\neq 0$ and set $A=\\nabla^{2}F_{1}(u,v)$ and $B=\\nabla^{2}F_{2}(u,v)$. Then $A,B\\succeq 0$ and $A+B$ has rank $1$. For positive semidefinite matrices, $\\mathrm{range}(A)\\subseteq \\mathrm{range}(A+B)$ and $\\mathrm{range}(B)\\subseteq \\mathrm{range}(A+B)$, so the range of $A$ and $B$ is in the same one-dimensional subspace. Hence there exists $\\theta_{u,v}\\in[0,1]$ such that \\[ A=\\theta_{u,v}(A+B). \\] Comparing the $(2,2)$ entries gives $\\varphi_1(u)=\\theta_{u,v}\\varphi(u)$, so $\\theta_{u,v}$ depends only on $u$, say $\\theta_{u,v}=\\theta_u$. Comparing the $(1,2)$ entries gives $\\varphi_1'(u)=\\theta_u\\varphi'(u)$. Since $\\varphi_1=\\theta\\varphi$, differentiating yields $\\varphi_1'=\\theta'\\varphi+\\theta \\varphi'$, hence $\\theta'(u)\\varphi(u)=0$ and $\\theta'(u)=0$. Hence $\\theta$ is constant. Finally $\\varphi_1=\\theta\\varphi$ and $\\varphi_2=(1-\\theta)\\varphi$, hence $\\varphi_1$ and $\\varphi_2$ are colinear, thus $\\varphi$ is extreme.<\/p>\n<p style=\"text-align:justify;\"><strong>Solving the extremality ODE $Q=0$.<\/strong> Let $\\varphi\\in\\mathcal{K}''\\setminus\\{0\\}$. Assume that $Q=0$. Then the function $\\psi=1\/\\varphi$ satisfies $\\psi''(u)=-Q(u)\/\\varphi(u)^3=0$, hence $\\psi(u)=\\alpha u+\\beta$, and \\[ \\varphi(u)=\\frac{1}{\\alpha u+\\beta}\\quad\\text{with}\\quad\\alpha u+\\beta > 0\\text{ on }\\mathbb{R}_{+}. \\] Up to multiplication by a positive constant, this yields<\/p>\n<p style=\"text-align:justify;\">\n<ul>\n<li>$\\varphi(u)\\equiv c$ with $c > 0$ <\/li>\n<li>$\\varphi(u)=\\frac{c}{u+t}$ with $c > 0$ and $u+t > 0$ on $\\mathbb{R}_+$. <\/li>\n<\/ul>\n<p style=\"text-align:justify;\">Back to $\\mathcal{K}$ and $\\Phi$, this gives, recalling that $\\varphi=\\Phi''$,<\/p>\n<p style=\"text-align:justify;\">\n<ul>\n<li>If $\\Phi''(u)=c$, then $\\Phi(u)=\\frac{c}{2}u^2+\\text{affine}$. <\/li>\n<li>If $\\Phi''(u)=\\frac{c}{u+t}$, then $\\Phi(u)=c(u+t)\\log(u+t)+\\text{affine}$. <\/li>\n<\/ul>\n<p style=\"text-align:justify;\">Note that if we define $H(x):=x\\log x-x$, $x > 0$, then for all $t\\geq0$, the function \\begin{align*}   K_t(u)&:=H(u+t)-H(1+t)-(u-1)H'(1+t)\\\\         &=(u+t)\\log(u+t)\\underbrace{-(u+t)\\log(1+t)-u+1}_{\\mathrm{affine}}\\\\         &= (u+t)\\log\\frac{u+t}{1+t}-(u-1), \\end{align*} $u > 0$, satisfies $K_t(1)=K_t'(1)=0$ and \\[   K_t''(u)=\\frac{1}{u+t},   \\quad K_{t}'''(u)=-\\frac{1}{(u+t)^2},   \\quad K_{t}''''(u)=\\frac{2}{(u+t)^3}. \\]<\/p>\n<p style=\"text-align:justify;\"><strong>Stieltjes-Nevanlinna classical theorem.<\/strong> This theorem (recalled here without proof) states that for all $\\varphi:(0,\\infty)\\to[0,\\infty)$, the following two properties are equivalent:<\/p>\n<p style=\"text-align:justify;\">\n<ul>\n<li>There exist a constant $c\\geq0$ and a positive Borel measure $\\nu$ on   $[0,\\infty)$ such that   \\[     \\int\\frac{1}{1+t}\\mathrm{d}\\nu(t) < \\infty     \\quad\\text{and}\\quad     \\varphi(u)=c+\\int\\frac{1}{u+t}\\mathrm{d}\\nu(t)\\quad\\text{for all }u>0.   \\] <\/li>\n<li>$\\varphi$ extends to a holomorphic function on   $\\mathbb{C}\\setminus(-\\infty,0]$, satisfies   \\[     \\varphi(x)\\geq0\\text{ for }x > 0,\\quad     \\Im \\varphi(z)\\leq 0\\text{ for }\\Im z>0,     \\quad\\text{and }     \\lim_{x\\to+\\infty} \\varphi(x)\\text{ exists in }[0,\\infty).   \\] <\/li>\n<\/ul>\n<p style=\"text-align:justify;\">In either case, $c=\\lim_{x\\to+\\infty} \\varphi(x)$ and $\\nu$ is uniquely determined by $\\varphi$.<\/p>\n<p style=\"text-align:justify;\">We say that such a $\\varphi$ is a <em>Stieltjes function<\/em>. Such a function is always $C^{\\infty}$ on $(0,\\infty)$ and <em>completely monotone<\/em> : $(-1)^{n}\\varphi^{(n)}(u)\\geq 0$ for all $n\\geq 0$ and $u > 0$.<\/p>\n<p style=\"text-align:justify;\"><strong>Integral representation of Stieltjes extremes.<\/strong> If $\\Phi$ is $\\mathcal{C}^4(\\mathbb{R}_+)$ and $\\varphi=\\Phi''$ is a Stieltjes function, with integral representation \\[ \\varphi(u)=c+\\int\\frac{1}{u+t}\\mathrm{d}\\nu(t), \\quad u > 0, \\] then $\\Phi\\in\\mathcal{K}$ and there exists an affine function $\\ell$ such that for all $u > 0$, \\[   \\Phi(u)=\\ell(u)+R(u)   \\quad\\text{with}\\quad R(u):=\\frac{c}{2}u^2+\\int K_t(u)\\mathrm{d}\\nu(t). \\]<\/p>\n<p style=\"text-align:justify;\"><strong>Proof.<\/strong> Since $K_t''(u)=1\/(u+t)$, we have \\[ \\frac{\\mathrm{d}^2}{\\mathrm{d}u^2}\\left(\\frac{c}{2}u^2+\\int K_{t}(u)\\mathrm{d}\\nu(t)\\right) =c+\\int\\frac{1}{u+t}\\mathrm{d}\\nu(t)=\\varphi(u). \\] The differentiation under the integral is valid by dominated convergence thanks to the integrability properties of $\\nu$, more precisely $K_t''(u)=1\/(u+t)\\leq\\max(1,1\/u)\/(1+t)$ and $1\/(u+t)^k\\leq u^{1-k}\/(u+t)$ for $k=2$ and $k=3$. Choosing the affine function $\\ell(u)=(\\Phi(1)-R(1))+(\\Phi'(1)-R'(1))(u-1)$ yields the identity. Finally, for $u > 0$, we have \\[ \\varphi'(u)=-\\int\\frac{1}{(u+t)^2}\\mathrm{d}\\nu(t), \\quad\\text{and}\\quad \\varphi''(u)=2\\int\\frac{1}{(u+t)^3}\\mathrm{d}\\nu(t). \\] By Cauchy-Schwarz in $L^2(\\nu)$ applied to $(u+t)^{-1\/2}$ and $(u+t)^{-3\/2}$, \\[ \\left(\\int\\frac{1}{(u+t)^{2}}\\mathrm{d}\\nu(t)\\right)^2 \\le \\left(\\int\\frac{1}{u+t}\\mathrm{d}\\nu(t)\\right) \\left(\\int\\frac{1}{(u+t)^3}\\mathrm{d}\\nu(t)\\right), \\] therefore \\[ \\varphi(u)\\varphi''(u)-2\\varphi'(u)^2 \\geq 2c\\int\\frac{1}{(u+t)^3}\\mathrm{d}\\nu(t)\\geq 0 \\] so $(u,v)\\mapsto\\varphi(u)v^2$ is convex and $\\Phi\\in\\mathcal{K}$.<\/p>\n<p style=\"text-align:justify;\"><strong>Counter examples.<\/strong> For a non-affine $\\Phi$, the condition of being in $\\mathcal{K}$ is equivalent to concavity of $\\psi:=1\/\\Phi''$ on $\\mathbb{R}_+$, which is strictly weaker than the Stieltjes assumption (analytic continuation plus a half-plane sign condition). In other words, beyond Stieltjes functions, there exist non-affine elements of $\\mathcal{K}$ that are not mixtures of shifts of $u\\mapsto u\\log(u)$. Let us give explicit counterexamples.<\/p>\n<p style=\"text-align:justify;\"><em>Our first counter example is $C^{\\infty}$<\/em>. Let $h\\in C^\\infty(\\mathbb{R}_+)$, $h\\neq0$, $h\\geq0$, with compact support included in the interval $(1,2)$. Fix $\\varepsilon > 0$ and define \\[ g(u):=1+u-\\varepsilon\\int_0^u(u-s)h(s)\\mathrm{d}s. \\] Then $g\\in C^\\infty(\\mathbb{R}_+)$ and $g$ is concave since $g''(u)=-\\varepsilon h(u)\\leq 0$. Moreover $g > 0$ on $\\mathbb{R}_+$ for $\\varepsilon$ small enough. Next we define \\[ \\varphi(u)=\\Phi''(u)=\\frac{1}{g(u)}, \\] and we choose $\\Phi(1)$ and $\\Phi'(1)$ arbitrarily, integrating twice to obtain $\\Phi\\in C^\\infty(\\mathbb{R}_+)$. Now, since $1\/\\Phi''=g$ is concave, $\\Phi\\in\\mathcal{K}$. However $g''$ vanishes identically on $(0,a)$, $a:=\\inf(\\mathrm{supp}\\,h)\\in(1,2)$, but is not identically zero on $(a,2)$, therefore $g$ is not real-analytic at $u=a$. Indeed, since $g''\\equiv0$ on $(0,a)$, all derivatives of $g''$ at $a$ vanish. If $g''$ were real-analytic at $a$, it would vanish in a neighborhood of $a$, contradicting the definition of $a$. In particular, $\\Phi''$ cannot be a Stieltjes function, since Stieltjes functions extend holomorphically to $\\mathbb{C}\\setminus(-\\infty,0]$ and are therefore real-analytic on $(0,\\infty)$.<\/p>\n<p style=\"text-align:justify;\"><em>Our second counter example is real-analytic.<\/em> Let $h(u)=(1+u)^2\\mathrm{e}^{-u}$, $u\\geq0$, which is real-analytic and non-negative. Let $p > 0$ and $\\varepsilon > 0$ and define \\[ g(u)=1+pu-\\varepsilon\\int_{0}^{u}(u-s)h(s)\\mathrm{d}s. \\] Then $g$ is real-analytic on $\\mathbb{R}_+$ and $g''(u)=-\\varepsilon h(u)\\leq 0$ so $g$ is concave. Moreover, \\[ \\int_0^\\infty h(s)\\mathrm{d}s = \\int_0^\\infty(1+s)^2\\mathrm{e}^{-s}\\mathrm{d}s = 5, \\] so for any choice of parameters satisfying $\\varepsilon< p\/5$ we have, for all $u > 0$, \\[ g(u)\\geq 1+pu-\\varepsilon u\\int_0^\\infty h(s)\\mathrm{d}s = 1 + (p-5\\varepsilon)u>0. \\] Define $\\varphi(u)=\\Phi''(u)=1\/g(u)$, $u > 0$, and integrate twice. Then $\\Phi$ is real-analytic and belongs to $\\mathcal{K}$ since $1\/\\Phi''=g$ is concave. Let us show now that $\\Phi''$ is not a Stieltjes function for a concrete choice of parameters. Suppose that $\\varphi$ is a Stieltjes function. Then it is completely monotone, in particular $\\varphi'''(u)\\leq 0$ for all $u > 0$. We have, for any smooth positive $g$, \\[ \\varphi'''(u)=\\frac{-g(u)^2g'''(u)+6g(u)g'(u)g''(u)-6(g'(u))^3}{g(u)^4}. \\] As $u\\searrow0$ we have $g(0^+)=1$, $g'(0^+)=p$, $g''(0^+)=-\\varepsilon h(0)=-\\varepsilon$, and \\[   g'''(0^+)=-\\varepsilon h'(0^+)\\quad   h'(0^+)=\\left.\\frac{\\mathrm{d}}{\\mathrm{d}u}\\bigl((1+u)^2\\mathrm{e}^{-u}\\bigr)\\right|_{u=0}=1, \\] so $g'''(0^+)=-\\varepsilon$. Plugging into the numerator yields \\[ -\\varphi'''(0^+)g(0^+)^4 = -\\bigl(\\varepsilon(1-6p)-6p^3\\bigr) = -\\varepsilon(1-6p)+6p^3. \\] The right hand side is $ < 0$ for instance if $p=10^{-2}$ and $\\varepsilon=10^{-3}$, hence $\\varphi'''(0^+) > 0$. By continuity of $\\varphi'''$ as $u\\searrow0$, $\\varphi'''(u) > 0$ for all $u > 0$ small enough, contradicting complete monotonicity. In particular, $\\Phi$ cannot admit the shifted-$u\\log u$ representation.<\/p>\n<p style=\"text-align:justify;\"><strong>Personal.<\/strong> The idea of studying the integral representation of the convex cone of $\\Phi$-entropies in relation with the extremes $u\\mapsto u^2$ and $u\\mapsto u\\log(u)$ is already suggested in my 2004 (J. Kyoto) and 2006 (ESAIM) papers. It is the motivation of this post.<\/p>\n<p style=\"text-align:justify;\"><strong>Further reading.<\/strong> The notion of $\\Phi$-entropy was generalized to matrices and operators using traces by Joel Tropp and Richard Y. Chen (2014), and a study of the associated convex cone was then conducted by Frank Hansen and Zhihua Zhang (2015).<\/p>\n<p style=\"text-align:justify;\">\n<ul>\n<li>On this blog<br \/>   <a href=\"https:\/\/djalil.chafai.net\/blog\/2026\/01\/23\/herglotz-and-nevanlinna-integral-representations\/\">Herglotz and Nevanlinna integral representation<\/a><br \/>   (2026-01-23) <\/li>\n<li>On this blog<br \/>   <a href=\"https:\/\/djalil.chafai.net\/blog\/2025\/01\/25\/about-variance-and-entropy\/\">About variance and entropy<\/a><br \/>   (2025-01-25) <\/li>\n<li>Joel A. Tropp and Richard Yuhua Chen<br \/>   <strong>Subadditivity of matrix $\\varphi$-entropy and concentration of random matrices<\/strong><br \/>   Electronic Journal of Probability 19(27) 1-30 (2014) <\/li>\n<li>Frank Hansen and Zhihua Zhang<br \/>   <strong>Characterisation of matrix entropies<\/strong><br \/>   Letters in Mathematical Physics 105(10) 1399-1411 (2015) <\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>This post is about the structure of the set of $\\Phi$ for which the $\\Phi$-entropy is convex. It is a follow-up of a previous post&#8230;<\/p>\n<div class=\"more-link-wrapper\"><a class=\"more-link\" href=\"https:\/\/djalil.chafai.net\/blog\/2026\/03\/03\/convexity-of-entropies-and-integral-representation\/\">Continue reading<span class=\"screen-reader-text\">Convexity of entropies and integral representation<\/span><\/a><\/div>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":68},"categories":[1],"tags":[],"_links":{"self":[{"href":"https:\/\/djalil.chafai.net\/blog\/wp-json\/wp\/v2\/posts\/22447"}],"collection":[{"href":"https:\/\/djalil.chafai.net\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/djalil.chafai.net\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/djalil.chafai.net\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/djalil.chafai.net\/blog\/wp-json\/wp\/v2\/comments?post=22447"}],"version-history":[{"count":4,"href":"https:\/\/djalil.chafai.net\/blog\/wp-json\/wp\/v2\/posts\/22447\/revisions"}],"predecessor-version":[{"id":22463,"href":"https:\/\/djalil.chafai.net\/blog\/wp-json\/wp\/v2\/posts\/22447\/revisions\/22463"}],"wp:attachment":[{"href":"https:\/\/djalil.chafai.net\/blog\/wp-json\/wp\/v2\/media?parent=22447"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/djalil.chafai.net\/blog\/wp-json\/wp\/v2\/categories?post=22447"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/djalil.chafai.net\/blog\/wp-json\/wp\/v2\/tags?post=22447"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}