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MATHEMATICS
Ts. Ando (Tsuyoshi Andô)
On Some Classes of Convex Functions
(Presented by Academician P. S. Aleksandrov on 1 X 1960)
In the first part of the monograph by M. A. Krasnosel’skii and Ya. B. Rutickii \((^1)\), in connection with the theory of Orlicz spaces, classes of convex \(N\)-functions satisfying the so-called \(\Delta_2\)-, \(\Delta'\)-, \(\Delta_3\)-, \(\Delta^2\)-, and \(\Delta_\Phi\)-conditions are considered (definitions of these concepts are given below). In \((^1)\) various criteria are established for an \(N\)-function \(M(u)\) to belong to one of these classes. In the present work, necessary and sufficient conditions are established for an \(N\)-function to belong to the classes under consideration, formulated in terms of properties of the complementary function.
We first recall some general facts from the theory of \(N\)-functions (see \((^1)\)).
Let \(p(t)\) \((0 \le t < \infty)\) be a nondecreasing, right-continuous function satisfying the conditions: \(p(+0)=0\), \(p(t)>0\) for \(t>0\), and \(p(\infty)=\infty\). The convex function
\[ M(u)=\int_0^{|u|} p(t)\,dt \]
is called an \(N\)-function. This function has the following properties: it is even, positive for \(u\ne0\), \(M(u)/u\) is strictly increasing for \(u>0\), and
\[ \frac{1}{2}up\left(\frac{1}{2}u\right)\le M(u)\le up(u). \tag{1} \]
Put \(q(s)=\sup_{p(t)\le s} t\). This function has the same properties as the function \(p(t)\). The convex function
\[ N(v)=\int_0^{|v|} q(s)\,ds \]
is called the \(N\)-function complementary to \(M(u)\).
If the function \(p(t)\) is strictly increasing and continuous, then
\[ p[q(v)]=v,\qquad q[p(u)]=u. \tag{2} \]
Two \(N\)-functions \(M_1(u)\) and \(M_2(u)\) are called equivalent (this notion of equivalence was introduced in \((^1)\)) if, for large values of \(u\), the inequalities
\[ M_1(k_1u)\le M_2(u)\le M_1(k_2u), \tag{3} \]
hold, where \(k_1,k_2\) are positive numbers.
In (¹) it was proved that the function
\[ M_1(u)=\int_0^{|u|}\frac{M(t)}{t}\,dt \tag{4} \]
is an \(N\)-function equivalent to \(M(u)\).
Theorem. In order that an \(N\)-function \(M(u)\) satisfy one of the conditions (the \(\Delta\)-conditions) in the first column, it is necessary and sufficient that the complementary \(N\)-function \(N(v)\) satisfy the corresponding condition (the \(\nabla\)-condition) in the second column:
\[ \begin{array}{ll} \text{for large } u,v & \text{for large } u,v \\[2mm] (\Delta_2):\quad M(2u)\leq kM(u) & (\nabla_2):\quad 2kN(u)\leq N(ku) \\[1mm] (\Delta'):\quad M(uv)\leq kM(u)M(v) & (\nabla'):\quad N(u)N(v)\leq N(kuv) \\[1mm] (\Delta_3):\quad uM(u)\leq M(ku) & (\nabla_3):\quad uN[N(u)]\leq kN^2(u) \\[1mm] (\Delta^2):\quad M^2(u)\leq M(ku) & (\nabla^2):\quad N(u^2)\leq kuN(u) \\[2mm] (\Delta_\Phi):\quad \Phi[M(u)]\leq M(ku) & (\nabla_\Phi):\quad \dfrac{N^{-1}(u)}{u}\leq \dfrac{kN^{-1}[\Phi(u)]}{\Phi(u)} \\[4mm] (\Delta_\Phi^*):\quad \Phi\!\left[\dfrac{M(u)}{u}\right]\leq \dfrac{M(ku)}{u} & (\nabla_\Phi^*):\quad N[\Phi(u)]\leq k\,\dfrac{\Phi(u)N(u)}{u} \end{array} \]
In the last two lines, \(\Phi(u)\) denotes some \(N\)-function.
The assertions of the theorem for the cases of the \(\Delta_2\)- and \(\Delta^2\)-conditions were proved in (¹). We shall consider the remaining cases. First of all, note that in every class of equivalent \(N\)-functions there is a function with a strictly increasing continuous derivative (for example, the \(N\)-function (4)). Therefore, without loss of generality, we may assume that the derivatives \(p(u)\) and \(q(u)\) considered below are continuous and strictly increasing.
Lemma 1. In order that an \(N\)-function \(M(u)\) satisfy the \(\Delta'\)-condition, it is necessary and sufficient that its derivative \(p(u)\) satisfy the analogous condition: for some \(k'\) and \(u_0>0\),
\[ p(uv)\leq k'p(u)p(v)\qquad (u,v\geq u_0). \tag{5} \]
Proof. Let
\[ M(uv)\leq kM(u)M(v)\qquad (u,v\geq u_1). \tag{\(\Delta'\)} \]
Then, by virtue of (1) and the fact that \(M(u)\) satisfies the \(\Delta_2\)-condition,
\[ p(uv)\leq \frac{M(2uv)}{uv}\leq k_1\frac{M(uv)}{uv}\qquad (u,v\geq u_2). \]
By virtue of \((\Delta')\) and (1),
\[ p(uv)\leq kk_1\frac{M(u)M(v)}{uv}\leq kk_1p(u)p(v)\qquad (u,v\geq \max\{u_1,u_2\}). \]
The necessity of the condition of the lemma is proved. Sufficiency is proved analogously. The lemma is proved.
It follows from (2) that \(p(u)\) satisfies (5) if and only if the inverse function \(q(v)\) satisfies the condition
\[ q(u)q(v)\leq q(k'uv)\qquad (u,v\geq v_0=p(u_0)). \tag{6} \]
In the same way as Lemma 1, one proves:
Lemma 2. In order that \(N(v)\) satisfy the \(\nabla'\)-condition, it is necessary and sufficient that its derivative \(p(v)\) satisfy condition (6).
Combining Lemmas 1 and 2, we obtain the assertion of the theorem for the case of the \(\Delta'\)-condition. This part of the theorem was proved in [1] under the additional assumption that the \(N\)-function \(N(v)\) satisfies the \(\Delta_3\)-condition.
Let us prove the assertion of the theorem for the case of the \(\Delta_\Phi\)-condition. Suppose that, for \(u \geqslant u_0\),
\[ \Phi[M(u)] \leqslant M(ku). \tag{\(\Delta_\Phi\)} \]
Let \(\varphi(u)\) be the derivative of \(\Phi(u)\). Then, by (1),
\[ p(u)\varphi[up(u)] \leqslant \frac{M(2u)}{u}\varphi[M(2u)] \leqslant \frac{\Phi[M(4u)]}{u} \leqslant \frac{M(4ku)}{u} \leqslant 4kp(4ku). \]
Putting \(k' = 4k\) and \(v = p(u)\), we obtain, by (2),
\[ v\varphi[vq(v)] \leqslant k'p[k'q(v)] \quad (v \geqslant v_0 = p(u_0)). \tag{7} \]
The function \(M(u)\), satisfying the \(\Delta_\Phi\)-condition, evidently also satisfies the \(\nabla_2\)-condition. Therefore its derivative, as is easy to show, satisfies, for large \(u\), the inequality
\[ 2p(u) \leqslant p(k''u). \tag{8} \]
It follows from (7) and (8) that, for some \(k_1\) and \(v_1\),
\[ v\varphi[vq(v)] \leqslant p[k_1q(v)] \quad (v \geqslant v_1), \]
i.e.,
\[ q\{v\varphi[vq(v)]\} \leqslant k_1q(v). \]
By (1),
\[ q\left\{\frac{\Phi[N(v)]}{q(v)}\right\} \leqslant k_1q(v). \]
Consequently,
\[ N\left\{\frac{\Phi[N(v)]}{q(v)}\right\} \leqslant k_1\Phi[N(v)], \]
so that
\[ N\left\{\frac{v\Phi[N(v)]}{N(v)}\right\} \leqslant k_1\Phi[N(v)] \quad (v \geqslant v_1). \]
Putting \(u = N(v)\), we obtain
\[ N\left[\frac{N^{-1}(u)\Phi(u)}{u}\right] \leqslant k_1\Phi(u) \quad (u \geqslant u_1 = N(v_1)), \]
whence \((\nabla_\Phi)\) follows.
Similarly, one proves that \((\Delta_\Phi)\) follows from \((\nabla_\Phi)\), and the other assertions of the theorem are proved.
In [1] the question was raised whether in every class of equivalent \(N\)-functions satisfying the \(\Delta'\)-condition there exists a function which satisfies this condition for all \(u\) and \(v\). This question has a negative answer. Let
\[ M(u) = (1+u)\ln(1+u) - u \quad (u \geqslant 0). \]
It is known that \(M(u)\) satisfies the \(\Delta'\)-condition. Suppose that there exists an \(N\)-function \(\Phi(u)\) equivalent to it such that
\[ \Phi(uv) \leqslant k\Phi(u)\Phi(v) \quad (u, v \geqslant 0). \tag{9} \]
By the equivalence of \(\Phi(u)\) and \(M(u)\), there exist \(k_1, k_2\) and \(u_0 > 0\) such that
\[ M(k_1u) \leqslant \Phi(u) \leqslant M(k_2u) \quad (u \geqslant u_0). \]
For any fixed \(u>0\) and \(v\geqslant \max \left\{u_0,\dfrac{u_0}{u}\right\}\),
\[ \Phi(uv)\geqslant M(k_1uv)=(1+k_1uv)\ln(1+k_1uv)-k_1uv, \]
\[ \Phi(v)\leqslant M(k_2v)=(1+k_2v)\ln(1+k_2v)-k_2v. \]
Therefore
\[ \frac{\Phi(uv)}{\Phi(v)}\geqslant \frac{(1+k_1uv)\ln(1+k_1uv)-k_1uv} {(1+k_2v)\ln(1+k_2v)-k_2v}, \]
whence it follows that
\[ \lim_{v\to\infty}\frac{\Phi(uv)}{\Phi(v)}\geqslant \frac{k_1}{k_2}\,u \qquad (u>0). \]
On the other hand,
\[ \overline{\lim_{v\to\infty}}\frac{\Phi(uv)}{\Phi(v)} \leqslant \overline{\lim_{v\to\infty}}\frac{k\Phi(u)\Phi(v)}{\Phi(v)} =k\Phi(u). \]
Thus,
\[ \Phi(u)\geqslant \frac{k_1}{kk_2}\,u \]
for all \(u>0\), which contradicts the properties of the \(N\)-function \(\Phi(u)\) (from (1) it follows that \(\lim\limits_{u\to 0}\dfrac{\Phi(u)}{u}=0\)).
In (1) it is proved that the \(N\)-functions \(N(v)\) complementary to the \(N\)-functions \(M(u)\) satisfying the \(\Delta_3\)- (respectively, \(\Delta^2\)-) condition satisfy the \(\Delta_2\)- (respectively, \(\Delta'\)-) condition. These assertions follow in an obvious way from the theorem proved above.
Let us give an example of a function \(M(u)\) satisfying the \(\Delta_3\)-condition, but not satisfying the \(\nabla'\)-condition. Let
\[ p(t)= \begin{cases} t, & \text{for } 0\leqslant t\leqslant 2,\\ (nt)^{2(n+1)^2}, & \text{for } 2^{2^n}\leqslant t<2^{2^{n+1}}. \end{cases} \]
The \(N\)-function
\[ M(u)=\int_0^{|u|} p(t)\,dt \]
has the stated properties.
Hokkaido University
Sapporo, Japan
Received
30 IX 1960
REFERENCES
- M. A. Krasnosel’skii, Ya. B. Rutitskii, Convex Functions and Orlicz Spaces, 1958.