On an inequality between mean values

Tiberiu Popoviciu
(original), Inequalities, Mathematical Analysis, mean value results, paper

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Tiberiu Popoviciu
(Institutul de Calcul)

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Sur une inégalité entre des valeurs moyennes

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T. Popoviciu, Sur une inégalité entre des valeurs moyennes, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz., no. 381-409 (1972), pp. 1-8 (in French)

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1972 b -Popoviciu- Publ. Elektrotehn. Fak. Univ. Beograd – Sur une inegalite entre de valeurs moyennes

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1972 b -Popoviciu- Publ. Elektrotehn. Fak. Univ. Beograd - On an inequality between mean values
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381. ON AN INEQUALITY BETWEEN AVERAGE VALUES*

Tiberiu Popoviciu

  1. If we consider the development
(1) 1 ( 1 − has 1 x ) ( 1 − has 2 x ) ⋯ ( 1 − has n x ) = ∑ v = 0 + ∞ ( v + n − 1 n − 1 ) q v x v (1) 1 1 − has 1 x 1 − has 2 x ⋯ 1 − has n x = ∑ v = 0 + ∞   ( v + n − 1 n − 1 ) q v x v {:(1)(1)/((1-a_(1)x)(1-a_(2)x)cdots(1-a_(n)x))=sum_(v=0)^(+oo)((v+n-1)/(n-1))q_(v)x^(v):}\begin{equation*} \frac{1}{\left(1-a_{1} x\right)\left(1-a_{2} x\right) \cdots\left(1-a_{n} x\right)}=\sum_{v=0}^{+\infty}\binom{v+n-1}{n-1} q_{v} x^{v} \tag{1} \end{equation*}(1)1(1−has1x)(1−has2x)⋯(1−hasnx)=∑v=0+∞(v+n−1n−1)qvxv
and if the numbers has 1 , has 2 , … , has n has 1 , has 2 , … , has n a_(1),a_(2),dots,a_(n)a_{1}, a_{2}, ..., a_{n}has1,has2,…,hasnare non-negative, we have inequalities
(2) q v 2 ≦ q v − 1 q v + 1 ( v = 1 , 2 , … ; q 0 = 1 ) (2) q v 2 ≦ q v − 1 q v + 1 v = 1 , 2 , … ; q 0 = 1 {:(2)q_(v)^(2) <= q_(v-1)q_(v+1)(v=1,2,dots;q_(0)=1):}\begin{equation*} q_{v}{ }^{2} \leqq q_{v-1} q_{v+1}\left(v=1,2, \ldots; q_{0}=1\right) \tag{2} \end{equation*}(2)qv2≦qv−1qv+1(v=1,2,…;q0=1)
THE q v , ν = 1 , 2 , … q v , ν = 1 , 2 , … sqrt(q_(v)),nu=1,2,dots\sqrt{q_{v}}, \nu=1,2, \ldotsqv,ν=1,2,…are then average values ​​of the numbers has 1 , has 2 , … , has n has 1 , has 2 , … , has n a_(1),a_(2),dots,a_(n)a_{1}, a_{2}, ..., a_{n}has1,has2,…,hasnwho verify inequalities
(3) q 1 ≦ q 2 ≦ ⋯ ≦ q ν ≦ ⋯ (3) q 1 ≦ q 2 ≦ ⋯ ≦ q ν ≦ ⋯ {:(3)q_(1) <= sqrt(q_(2)) <= cdots <= sqrt(q_(nu)) <= cdots:}\begin{equation*} q_{1} \leqq \sqrt{q_{2}} \leqq \cdots \leqq \sqrt{q_{\nu}} \leqq \cdots \tag{3} \end{equation*}(3)q1≦q2≦⋯≦qν≦⋯
which can easily be deduced from (2).
Moreover, if n > 1 n > 1 n > 1n>1n>1and (non-negative) numbers has 1 , has 2 , … , has n has 1 , has 2 , … , has n a_(1),a_(2),dots,a_(n)a_{1}, a_{2}, ..., a_{n}has1,has2,…,hasnare not all equal; everywhere in (2) and (3) it is strict inequality (with the sign < < <<<) which takes place.
A proof of these properties can be found in the book by G.H. Hardy, J.E. Littlewood, and G. Pólya [2] (hereafter we will refer to these three authors as HLP). This proof amounts to expressing the coefficients of the expansion (1) calculated by the formula
(4) ( v + n − 1 n − 1 ) q v = [ has 1 , has 2 , … , has n ; x v + n − 1 ] (4) ( v + n − 1 n − 1 ) q v = has 1 , has 2 , … , has n ; x v + n − 1 {:(4)((v+n-1)/(n-1))q_(v)=[a_(1),a_(2),dots,a_(n);x^(v+n-1)]:}\begin{equation*} \binom{v+n-1}{n-1} q_{v}=\left[a_{1}, a_{2}, \ldots, a_{n}; x^{v+n-1}\right] \tag{4} \end{equation*}(4)(v+n−1n−1)qv=[has1,has2,…,hasn;xv+n−1]
the second member being the difference divided by order n − 1 n − 1 n-1n-1n−1on the nodes has 1 , has 2 , … , has n has 1 , has 2 , … , has n a_(1),a_(2),dots,a_(n)a_{1}, a_{2}, ..., a_{n}has1,has2,…,hasn(distinct or not) from the function x ν + n − 1 , ν = 0 , 1 , … x ν + n − 1 , ν = 0 , 1 , … x^(nu+n-1),nu=0,1,dotsx^{\nu+n-1}, \nu=0,1, \ldotsxν+n−1,ν=0,1,…We then make use of a well-known expression for the difference divided by a multiple integral due to A. Genocchi [1].
In the aforementioned book by HLP [2] (p. 164) the more general property that if n > 1 n > 1 n > 1n>1n>1and if the numbers has 1 , has 2 , … , has n has 1 , has 2 , … , has n a_(1),a_(2),dots,a_(n)a_{1}, a_{2}, ..., a_{n}has1,has2,…,hasnare real (not necessarily of the same sign) and not all equal, the quadratic form ∑ q μ + ν y μ y ν ∑ q μ + ν y μ y ν sumq_(mu+nu)y_(mu)y_(nu)\sum q_{\mu+\nu} y_{\mu} y_{\nu}∑qμ+νyμyνis positive and if the numbers has 1 , has 2 , … , has n has 1 , has 2 , … , has n a_(1),a_(2),dots,a_(n)a_{1}, a_{2}, ..., a_{n}has1,has2,…,hasnare non-negative, no
not all equal, the quadratic form ∑ q μ + ν + 1 y μ y ν ∑ q μ + ν + 1 y μ y ν sumq_(mu+nu+1)y_(mu)y_(nu)\sum q_{\mu+\nu+1} y_{\mu} y_{\nu}∑qμ+ν+1yμyνis positive. From the theory of definite quadratic forms, it follows that inequality (2) also remains true (with the restriction concerning the case of equality) when v v vvvis odd and has 1 , has 2 , … , has n has 1 , has 2 , … , has n a_(1),a_(2),dots,a_(n)a_{1}, a_{2}, ..., a_{n}has1,has2,…,hasnare real and arbitrary.
For the demonstration of inequalities (2), other integral representations of divided differences can also be used. We will return to this question at the end of this work.
2. HAS ` HAS ` A^(`)\grave{A}HAS`As an application of some more general results, I gave another proof of inequalities (2) and (3) [3]. At that time I was unaware of the book cited [2] by HLP*).
My proof also uses formula (4), but differs significantly from that given by HLP. It is based on the theory of higher-order convex functions. My research on linear functionals, which I have called simple, allows me to considerably generalize my results from the work cited in [3]. Before giving this generalization, I will recall the main properties, which will be used here, of higher-order convex functions and linear functionals of the simple form. For more details, the reader is asked to consult my previous works and especially work [4] in the bibliography. Work [5] can also be consulted.
3. Let I I IIIan interval (of non-zero length) of the real axis R R R\mathbf{R}RAnd m m mmman integer ≧ − 1 ≧ − 1 >= -1\geqq-1≧−1The function f : I → R f : I → R f:I rarrRf: I \rightarrow \mathbf{R}f:I→Ris said to be non-concave of order m m mmm(on I I III) if inequality
(5)
[ x 1 , x 2 , … , x m + 2 ; f ] ≧ 0 x 1 , x 2 , … , x m + 2 ; f ≧ 0 [x_(1),x_(2),dots,x_(m+2);f] >= 0\left[x_{1}, x_{2}, \ldots, x_{m+2}; f\right] \geqq 0[x1,x2,…,xm+2;f]≧0
is checked for any group of m + 2 m + 2 m+2m+2m+2distinct points x ν , ν = 1 , 2 , … , m + 2 x ν , ν = 1 , 2 , … , m + 2 x_(nu),nu=1,2,dots,m+2x_{\nu}, \nu=1,2, \ldots, m+2xν,ν=1,2,…,m+2of I I IIIThe function f f fffis said to be convex of order m m mmmif in (5) it is the strict inequality (with the sign > > >>>) which is always verified. The first member of (5) shows the divided difference (of order m + 1 m + 1 m+1m+1m+1) of the function f f fffon the points (or nodes) x v , v = 1 , 2 , … , m + 2 x v , v = 1 , 2 , … , m + 2 x_(v),v=1,2,dots,m+2x_{v}, v=1,2, \ldots, m+2xv,v=1,2,…,m+2.
Any convex function of order m m mmmis a non-concave function of order m m mmmWe define the differences divided over
nodes that are not necessarily distinct, as usual, using successive derivatives of the function. If then the function f f fffis non-concave of order m m mmmInequality (5) remains true regardless of the points x v , v = 1 , 2 , … , m + 2 x v , v = 1 , 2 , … , m + 2 x_(v),v=1,2,dots,m+2x_{v}, v=1,2, \ldots, m+2xv,v=1,2,…,m+2distinct or not. If the function f f fffis convex of order m m mmmand if m ≥ 0 m ≥ 0 m >= 0m \geq 0m≥0, strict inequality (with the sign > 0 > 0 > 0>0>0) remains true provided that the points x y x y x_(y)x_{\mathrm{y}}xynot all be confused. We assume, of course, that the left-hand side of (5) exists, therefore that this divided difference is defined in the way we have indicated (which implies the existence of certain derivatives of f f fff). For m = − 1 m = − 1 m=-1m=-1m=−1The property is trivial.
We know that a non-concave function of order m m mmmis continuous on the inside of I I IIIif m > 0 m > 0 m > 0m>0m>0and has a continuous derivative of order m − 1 m − 1 m-1m-1m−1on the inside of I I IIIif m > 1 m > 1 m > 1m>1m>1.
If the derivative f ( m + 1 ) f ( m + 1 ) f^((m+1))f^{(m+1)}f(m+1)order m + 1 m + 1 m+1m+1m+1exists ( f ( 0 ) = f ) f ( 0 ) = f (f^((0))=f)\left(f^{(0)}=f\right)(f(0)=f)the condition ∀ f ( m + 1 ) ( x ) ≧ 0 ∀ f ( m + 1 ) ( x ) ≧ 0 AAf^((m+1))(x) >= 0\forall f^{(m+1)}(x) \geqq 0∀f(m+1)(x)≧0is necessary and sufficient for non-concavity of order m m mmmand the condition
∀ f ( m + 1 ) ( x ) > 0 ∀ f ( m + 1 ) ( x ) > 0 AAf^((m+1))(x) > 0\forall f^{(m+1)}(x)>0∀f(m+1)(x)>0is sufficient for the convexity of order m m mmmof the function f f fff. When m ≧ 0 m ≧ 0 m >= 0m \geqq 0m≧0the derivative f ( m + 1 ) f ( m + 1 ) f^((m+1))f^{(m+1)}f(m+1)may cancel each other out on certain points of I I IIIfor a convex function of order m m mmmBut this derivative must then be different from zero on a set that is dense everywhere in I I III. If m ≧ 0 m ≧ 0 m >= 0m \geqq 0m≧0the conditions f ( m + 1 ) ( x ) ≧ 0 f ( m + 1 ) ( x ) ≧ 0 f^((m+1))(x) >= 0f^{(m+1)}(x) \geqq 0f(m+1)(x)≧0on I I IIIAnd f ( m + 1 ) ( x ) > 0 f ( m + 1 ) ( x ) > 0 f^((m+1))(x) > 0f^{(m+1)}(x)>0f(m+1)(x)>0on a dense set everywhere in I I IIIare necessary and sufficient for the convexity of order m m mmmof f f fffon I I IIIThis property stems from the fact that if for a function f f fffnon-concave of order m m mmmWe have [ x 1 , x 2 , … , x m + 2 ; f ] = 0 x 1 , x 2 , … , x m + 2 ; f = 0 [x_(1),x_(2),dots,x_(m+2);f]=0\left[x_{1}, x_{2}, \ldots, x_{m+2}; f\right]=0[x1,x2,…,xm+2;f]=0, this function reduces to a polynomial of degree m m mmmon the smallest closed interval containing the points x 1 , x 2 , … , x m + 2 x 1 , x 2 , … , x m + 2 x_(1),x_(2),dots,x_(m+2)x_{1}, x_{2}, \ldots, x_{m+2}x1,x2,…,xm+2and therefore has a derivative ( m + 1 ) th ( m + 1 ) th  (m+1)^("th ")(m+1)^{\text {th}}(m+1)th zero on the interior of this last interval (if its length is not zero). A consequence of this property will be applied in the form of
Lemma 1. If m ≧ 0 m ≧ 0 m >= 0m \geqq 0m≧0and the derivative of order m + 1 m + 1 m+1m+1m+1of the polynomial P P PPPis not identically zero and is non-negative on I, this polynomial is a convex function of order m m mmmon I I III.
Indeed P ( m + 1 ) P ( m + 1 ) P^((m+1))P^{(m+1)}P(m+1)can then only be zero at most on a finite number of points, therefore is different from zero on a set that is dense everywhere in I I III4. Now R [ f ] R [ f ] R[f]R[f]R[f]a linear functional (additive and homogeneous) defined on a linear set S S SSSfunctions f f fffreal, continuous and defined on the interval 1. The set S S SSScan be formed by all continuous functions defined on I I III, but also by only a part of these functions. We will assume that S S SSSalways contains all polynomials.
If the equalities
(6) R [ 1 ] = R [ x ] = ⋯ = R [ x m ] = 0 (6) R [ 1 ] = R [ x ] = ⋯ = R x m = 0 {:(6)R[1]=R[x]=cdots=R[x^(m)]=0:}\begin{equation*} R[1]=R[x]=\cdots=R\left[x^{m}\right]=0 \tag{6} \end{equation*}(6)R[1]=R[x]=⋯=R[xm]=0
as well as inequality
(7) R [ x m + 1 ] ≠ 0 (7) R x m + 1 ≠ 0 {:(7)R[x^(m+1)]!=0:}\begin{equation*} R\left[x^{m+1}\right] \neq 0 \tag{7} \end{equation*}(7)R[xm+1]≠0
are verified for a certain integer m ≧ − 1 m ≧ − 1 m >= -1m \geqq-1m≧−1, we say that the linear functional R [ f ] R [ f ] R[f]R[f]R[f]is the degree of accuracy m m mmm(or that m m mmm(is its degree of accuracy). This number m m mmmIf it exists, it is well-defined (is unique). When m = − 1 m = − 1 m=-1m=-1m=−1relations (6), (7) must be replaced by the single relation R [ 1 ] ≠ 0 R [ 1 ] ≠ 0 R[1]!=0R[1] \neq 0R[1]≠0.
Finally, we will recall the notion of a functional of simple form. The linear functional R [ f ] R [ f ] R[f]R[f]R[f]degree of accuracy m m mmmis said to be of the simple form if it enjoys the property that we have R [ f ] ≠ 0 R [ f ] ≠ 0 R[f]!=0R[f] \neq 0R[f]≠0for any function f f fffconvex of order m m mmm(on I). In this case, we have R [ f ] R [ x m + 1 ] > 0 R [ f ] R x m + 1 > 0 R[f]R[x^(m+1)] > 0R[f] R\left[x^{m+1}\right]>0R[f]R[xm+1]>0for any function f f fffconvex of order m m mmmIn what follows, we can only consider functionals of degree of exactness. m m mmmand of the simple form for which R [ x m + 1 ] > 0 R x m + 1 > 0 R[x^(m+1)] > 0R\left[x^{m+1}\right]>0R[xm+1]>0Such a linear functional satisfies the inequality R [ f ] > 0 R [ f ] > 0 R[f] > 0R[f]>0R[f]>0for any function f f fffconvex of order m m mmmOtherwise, the linear functional − R [ f ] − R [ f ] -R[f]-R[f]−R[f], which is also a degree of accuracy m m mmmand of the simple form, checks the property.
A linear functional R [ f ] R [ f ] R[f]R[f]R[f]of the simple form satisfies an important formula for the mean (see [4]) which we will not use in this work. Moreover, the existence of this formula for R [ f ] R [ f ] R[f]R[f]R[f]degree of accuracy m m mmmis precisely equivalent to the property that R [ f ] ≠ 0 R [ f ] ≠ 0 R[f]!=0R[f] \neq 0R[f]≠0for any function f f fffconvex of order m m mmm5.
Either R [ f ] R [ f ] R[f]R[f]R[f]a linear function of the previous form. The numbers
(8) c v = R [ x v ] , ν = 0 , 1 , … (8) c v = R x v , ν = 0 , 1 , … {:(8)c_(v)=R[x^(v)]","quad nu=0","1","dots:}\begin{equation*} c_{v}=R\left[x^{v}\right], \quad \nu=0,1, \ldots \tag{8} \end{equation*}(8)cv=R[xv],ν=0,1,…
are the moments of this function.
When R [ f ] R [ f ] R[f]R[f]R[f]is of the last degree of accuracy m m mmm, THE m + 1 m + 1 m+1m+1m+1, first moments are null (if m ≥ 0 m ≥ 0 m >= 0m \geq 0m≥0If we assume that R [ f ] R [ f ] R[f]R[f]R[f]either degree of accuracy m m mmmand in its simple form, it exists between moments c v , v = m + 1 , m + 2 , … c v , v = m + 1 , m + 2 , … c_(v),v=m+1,m+2,dotsc_{\mathrm{v}}, v=m+1, m+2, \ldotscv,v=m+1,m+2,…, certain inequalities that we will highlight. These inequalities can be deduced from the
Theorem 1. Let m ≥ 0 m ≥ 0 m >= 0m \geq 0m≥0And R [ f ] R [ f ] R[f]R[f]R[f]a linear functional defined on S S SSSdegree of accuracy m m mmmof the simple form and R [ x m + 1 ] > 0 R x m + 1 > 0 R[x^(m+1)] > 0R\left[x^{m+1}\right]>0R[xm+1]>0Let's ask.
(9) ( v + m + 1 m + 1 ) q v = c v + m + 1 ( v = 0 , 1 , … ) (9) ( v + m + 1 m + 1 ) q v = c v + m + 1 ( v = 0 , 1 , … ) {:(9)((v+m+1)/(m+1))q_(v)=c_(v+m+1)quad(v=0","1","dots):}\begin{equation*} \binom{v+m+1}{m+1} q_{v}=c_{v+m+1} \quad(v=0,1, \ldots) \tag{9} \end{equation*}(9)(v+m+1m+1)qv=cv+m+1(v=0,1,…)
and suppose that the polynomial ∑ v = 0 k a v x ν ∑ v = 0 k   a v x ν sum_(v=0)^(k)a_(v)x^(nu)\sum_{v=0}^{k} a_{v} x^{\nu}∑v=0khasvxνeither non-negative and not identically zero on the interval.
It follows that we have inequality
(10) ∑ v = 0 k a v q v = a 0 q 0 + a 1 q 1 + ⋯ + a k q k > 0 (10) ∑ v = 0 k   a v q v = a 0 q 0 + a 1 q 1 + ⋯ + a k q k > 0 {:(10)sum_(v=0)^(k)a_(v)q_(v)=a_(0)q_(0)+a_(1)q_(1)+cdots+a_(k)q_(k) > 0:}\begin{equation*} \sum_{v=0}^{k} a_{v} q_{v}=a_{0} q_{0}+a_{1} q_{1}+\cdots+a_{k} q_{k}>0 \tag{10} \end{equation*}(10)∑v=0khasvqv=has0q0+has1q1+⋯+haskqk>0
Indeed, the polynomial
P = ∑ v = 0 k a v x v + m + 1 ( v + 1 ) ( v + 2 ) ⋯ ( v + m + 1 ) 1 ( m + 1 ) ! ∑ v = 0 k a v x v + m + 1 ( v + m + 1 m + 1 ) P = ∑ v = 0 k   a v x v + m + 1 ( v + 1 ) ( v + 2 ) ⋯ ( v + m + 1 ) 1 ( m + 1 ) ! ∑ v = 0 k   a v x v + m + 1 ( v + m + 1 m + 1 ) P=sum_(v=0)^(k)(a_(v)x^(v+m+1))/((v+1)(v+2)cdots(v+m+1))(1)/((m+1)!)sum_(v=0)^(k)(a_(v)x^(v+m+1))/(((v+m+1)/(m+1)))P=\sum_{v=0}^{k} \frac{a_{v} x^{v+m+1}}{(v+1)(v+2) \cdots(v+m+1)} \frac{1}{(m+1)!} \sum_{v=0}^{k} \frac{a_{v} x^{v+m+1}}{\binom{v+m+1}{m+1}}P=∑v=0khasvxv+m+1(v+1)(v+2)⋯(v+m+1)1(m+1)!∑v=0khasvxv+m+1(v+m+1m+1)
is convex with order m m mmm, as a consequence of Lemma 1. Taking into account (9) it follows that
R [ P ] = 1 ( m + 1 ) ! ∑ v = 0 k a v q v > 0 , R [ P ] = 1 ( m + 1 ) ! ∑ v = 0 k   a v q v > 0 , R[P]=(1)/((m+1)!)sum_(v=0)^(k)a_(v)q_(v) > 0,R[P]=\frac{1}{(m+1)!} \sum_{v=0}^{k} a_{v} q_{v}>0,R[P]=1(m+1)!∑v=0khasvqv>0,
which is equivalent to inequality (10).
Corollary 1. If R [ f ] R [ f ] R[f]R[f]R[f]is a linear functional that satisfies the hypotheses of Theorem 1, the quadratic form in y 0 , y 1 , … , y r y 0 , y 1 , … , y r y_(0),y_(1),dots,y_(r)y_{0}, y_{1}, \ldots, y_{r}y0,y1,…,yr,
(11) ∑ μ , v = 0 r q μ + v + s y μ y v (11) ∑ μ , v = 0 r   q μ + v + s y μ y v {:(11)sum_(mu,v=0)^(r)q_(mu+v+s)y_(mu)y_(v):}\begin{equation*} \sum_{\mu, v=0}^{r} q_{\mu+v+s} y_{\mu} y_{v} \tag{11} \end{equation*}(11)∑μ,v=0rqμ+v+syμyv
is defined as positive, when:
1 ∘ s 1 ∘ s 1^(@)s1^{\circ} s1∘sis an even integer ≧ 0 ≧ 0 >= 0\geqq 0≧0And r r rrran integer ≧ 0 ≧ 0 >= 0\geqq 0≧0, Or
2 ∘ I 2 ∘ I 2^(@)I2^{\circ} I2∘Iis a positive interval and s , r s , r s,rs, rs,rare integers ≧ 0 ≧ 0 >= 0\geqq 0≧0We
will say that the interval I I IIIis positive if it contains no non-positive points, so if x ∈ I ⇒ x > 0 x ∈ I ⇒ x > 0 x in I=>x > 0x \in I \Rightarrow x>0x∈I⇒x>0.
For the demonstration, it suffices to note that the polynomial x s ( ∑ ν = 0 r y ν x ν ) 2 x s ∑ ν = 0 r   y ν x ν 2 x^(s)(sum_(nu=0)^(r)y_(nu)x^(nu))^(2)x^{s}\left(\sum_{\nu=0}^{r} y_{\nu} x^{\nu}\right)^{2}xs(∑ν=0ryνxν)2is non-negative on I I I\boldsymbol{I}I.
Property is equivalent to inequality
(12) | q s q s + 1 ⋯ q s + r q s + 1 q s + 2 q s + r + 1 ⋮ ⋮ q s + r q s + r + 1 q s + 2 r | > 0 ( s = 0 , 1 , … ) (12) q s q s + 1 ⋯ q s + r q s + 1 q s + 2 q s + r + 1 ⋮ ⋮ q s + r q s + r + 1 q s + 2 r > 0 ( s = 0 , 1 , … ) {:(12)|[q_(s),q_(s+1),cdots,q_(s+r)],[q_(s+1),q_(s+2),,q_(s+r+1)],[vdots,,,],[vdots,,,],[q_(s+r),q_(s+r+1),,q_(s+2r)]| > 0quad(s=0","1","dots):}\left|\begin{array}{cccc} q_{s} & q_{s+1} & \cdots & q_{s+r} \tag{12}\\ q_{s+1} & q_{s+2} & & q_{s+r+1} \\ \vdots & & & \\ \vdots & & & \\ q_{s+r} & q_{s+r+1} & & q_{s+2 r} \end{array}\right|>0 \quad(s=0,1, \ldots)(12)|qsqs+1⋯qs+rqs+1qs+2qs+r+1⋮⋮qs+rqs+r+1qs+2r|>0(s=0,1,…)
And s = 0 , 2 , 4 , … s = 0 , 2 , 4 , … s=0,2,4,dotss=0,2,4, \ldotss=0,2,4,…, respectively s = 0 , 1 , 2 , … s = 0 , 1 , 2 , … s=0,1,2,dotss=0,1,2, \ldotss=0,1,2,…
In particular, we have
(13) q s + 1 2 < q s q s + 2 (13) q s + 1 2 < q s q s + 2 {:(13)q_(s+1)^(2) < q_(s)q_(s+2):}\begin{equation*} q_{s+1}^{2}<q_{s} q_{s+2} \tag{13} \end{equation*}(13)qs+12<qsqs+2
for all integers s s ssschecking the reported restrictions.
6. Corollary 1 is obtained by particularizing the polynomial ∑ ν = 0 k a ν x ν ∑ ν = 0 k   a ν x ν sum_(nu=0)^(k)a_(nu)x^(nu)\sum_{\nu=0}^{k} a_{\nu} x^{\nu}∑ν=0khasνxνof theorem 1. We can obtain various inequalities of the same type by particularizing this polynomial in a different way.
If the interval I I IIIis bounded below and if a ≦ inf I a ≦ inf I a <= i n f Ia \leqq \inf Ihas≦infIthe polynomial
(14) ( x − a ) s ( ∑ v = 0 r y v x v ) 2 (14) ( x − a ) s ∑ v = 0 r   y v x v 2 {:(14)(x-a)^(s)(sum_(v=0)^(r)y_(v)x^(v))^(2):}\begin{equation*} (x-a)^{s}\left(\sum_{v=0}^{r} y_{v} x^{v}\right)^{2} \tag{14} \end{equation*}(14)(x−has)s(∑v=0ryvxv)2
is non-negative on I I IIIand we deduce that the quadratic form in y 0 , y 1 , … , y r ( q − a ) ( s ) ( ∑ v = 0 r q y v ) ( 2 ) y 0 , y 1 , … , y r ( q − a ) ( s ) ∑ v = 0 r   q y v ( 2 ) y_(0),y_(1),dots,y_(r)(q-a)^((s))(sum_(v=0)^(r)qy_(v))^((2))y_{0}, y_{1}, \ldots, y_{r}(q-a)^{(s)}\left(\sum_{v=0}^{r} q y_{v}\right)^{(2)}y0,y1,…,yr(q−has)(s)(∑v=0rqyv)(2)is positive. Here the exponents ( s s sss), (2) denote
usual symbolic powers. This amounts to ordering the polynomial (14) according to the powers of x = q x = q x=qx=qx=qand then to replace q ν q ν q^(nu)q^{\nu}qνby q ν q ν q_(nu)q_{\nu}qνThe number s s sssis any non-negative integer. The quadratic form enjoys the same property. ( q − b ) ( s ) ( ∑ ν = 0 r q y ν ) ( 2 ) ( q − b ) ( s ) ∑ ν = 0 r   q y ν ( 2 ) (q-b)^((s))(sum_(nu=0)^(r)qy_(nu))^((2))(q-b)^{(s)}\left(\sum_{\nu=0}^{r} q y_{\nu}\right)^{(2)}(q−b)(s)(∑ν=0rqyν)(2)if sup I ≦ b < + ∞ sup I ≦ b < + ∞ s u p I <= b < +oo\sup I \leqq b<+\inftysupI≦b<+∞These properties are equivalent
to the positivity of certain Hankel determinants, and therefore to inequalities analogous to (12). It is unnecessary to dwell on other special cases here.
7. By particularizing the linear functional R [ f ] R [ f ] R[f]R[f]R[f]We can obtain various particular inequalities, some more interesting than others.
Let a 1 , a 2 , … , a n , n > 1 a 1 , a 2 , … , a n , n > 1 a_(1),a_(2),dots,a_(n),n > 1a_{1}, a_{2}, \ldots, a_{n}, n>1has1,has2,…,hasn,n>1points not all coincide in the interval I. Then
(15) R [ f ] = [ a 1 , a 2 , … , a n ; f ] (15) R [ f ] = a 1 , a 2 , … , a n ; f {:(15)R[f]=[a_(1),a_(2),dots,a_(n);f]:}\begin{equation*} R[f]=\left[a_{1}, a_{2}, \ldots, a_{n} ; f\right] \tag{15} \end{equation*}(15)R[f]=[has1,has2,…,hasn;f]
is a linear functional of degree of accuracy n − 2 n − 2 n-2n-2n−2, of the simple form (ipso facto, according to the very definition of higher-order convex functions
) and which is well-defined on any polynomial. We can apply the preceding theory and easily recover the results of our earlier work [3] and, in particular, those presented in No. 1 of the present work concerning the coefficients of the expansion (1).
8. Suppose that I = [ a , b ] I = [ a , b ] I=[a,b]I=[a, b]I=[has,b]Let be a bounded and closed interval and consider the linear functional
(16) R [ f ] = ∫ a b φ f ( m + 1 ) d x ( m ≧ − 1 ) (16) R [ f ] = ∫ a b   φ f ( m + 1 ) d x ( m ≧ − 1 ) {:(16)R[f]=int_(a)^(b)varphif^((m+1))dx quad(m >= -1):}\begin{equation*} R[f]=\int_{a}^{b} \varphi f^{(m+1)} d x \quad(m \geqq-1) \tag{16} \end{equation*}(16)R[f]=∫hasbφf(m+1)dx(m≧−1)
Or φ φ varphi\varphiφis a continuous, non-negative function that is not identically zero on [ a , b ] [ a , b ] [a,b][a, b][has,b]Given that the values ​​of R [ f ] R [ f ] R[f]R[f]R[f]Since polynomials are our only interest at the moment, we can assume that S S SSSis formed by all the functions f f fffhaving a continuous derivative of order m + 1 ( f ( 0 ) = f ) m + 1 f ( 0 ) = f m+1(f^((0))=f)m+1\left(f^{(0)}=f\right)m+1(f(0)=f)on [ a , b ] [ a , b ] [a,b][a, b][has,b]In this case, (16) is indeed a linear functional of degree of accuracy m m mmmof the simple form and we also have R [ x m + 1 ] > 0 R x m + 1 > 0 R[x^(m+1)] > 0R\left[x^{m+1}\right]>0R[xm+1]>0(see [6]). Taking into account (8) and (9), it can be noted that the q ν ( m + 1 ) ! q ν ( m + 1 ) ! (q_(nu))/((m+1)!)\frac{q_{\nu}}{(m+1)!}qν(m+1)!are precisely the moments in the classical sense of the distribution function ∫ φ d x ∫ φ d x int varphi dx\int \varphi d x∫φdx.
Note that (15) is also of the form (16), the set S S SSSbeing defined as above. If, indeed, n > 1 n > 1 n > 1n>1n>1And − ∞ < a ≦ min ( a ν ) < max ( a ν ) ≦≦ b < + ∞ − ∞ < a ≦ min a ν < max a ν ≦≦ b < + ∞ -oo < a <= min(a_(nu)) < max(a_(nu))≦≦b < +oo-\infty<a \leqq \min \left(a_{\nu}\right)<\max \left(a_{\nu}\right) \leqq \leqq b<+\infty−∞<has≦min(hasν)<max(hasν)≦≦b<+∞we have
(17) [ a 1 , a 2 , … , a n ; f ] = ∫ a b φ f ( n − 1 ) d x (17) a 1 , a 2 , … , a n ; f = ∫ a b   φ f ( n − 1 ) d x {:(17)[a_(1),a_(2),dots,a_(n);f]=int_(a)^(b)varphif^((n-1))dx:}\begin{equation*} \left[a_{1}, a_{2}, \ldots, a_{n} ; f\right]=\int_{a}^{b} \varphi f^{(n-1)} d x \tag{17} \end{equation*}(17)[has1,has2,…,hasn;f]=∫hasbφf(n−1)dx
Or φ φ varphi\varphiφis a continuous non-negative function on [ a , b ] [ a , b ] [a,b][a, b][has,b]and independent of f f fff(a so-called "spline" function). This formula is well known (see, e.g., [5]) and can be used instead of A. Genocchi's representation [1], to demonstrate inequality (2).
We can obviously consider, instead of (16), the case of a more general linear functional of the form
(18) ∫ a b f ( n + 1 ) d V , (18) ∫ a b   f ( n + 1 ) d V , {:(18)int_(a)^(b)f^((n+1))dV",":}\begin{equation*} \int_{a}^{b} f^{(n+1)} d V, \tag{18} \end{equation*}(18)∫hasbf(n+1)dV,
Or a , b ( a < b ) a , b ( a < b ) a,b(a < b)a, b(a<b)has,b(has<b)are the extremities (finite or not) of the (closed) interval I I IIIAnd V V VVVis a suitably chosen bounded-variation function.
Examples of linear functionals of simple form, which are generally of type (16), can be found in the theory of approximate quadrature formalisms. Thus, the remaining classical formulas of Côtes and Gauss, and many others, are such functionals.
If R [ f ] R [ f ] R[f]R[f]R[f]has a representation of the form (18) the preceding generally amounts to well-known properties of classical moment theory (de Stieltjes, Hamburger, Hausdorff, etc.). But our results are independent of any integral representation of R [ f ] R [ f ] R[f]R[f]R[f]9.
Under certain conditions, inequalities can be obtained between generalized moments.
Let's always apply that R [ f ] R [ f ] R[f]R[f]R[f]is a linear functional of degree of accuracy m ≧ 0 m ≧ 0 m >= 0m \geqq 0m≧0of the forma simole and its set of definition S S SSScontains the functions φ ν , ν = 0 , 1 , … , k φ ν , ν = 0 , 1 , … , k varphi_(nu),nu=0,1,dots,k\varphi_{\nu}, \nu=0,1, \ldots, kφν,ν=0,1,…,kcontinuous and having derivatives φ v ( m + 1 ) φ v ( m + 1 ) varphi_(v)^((m+1))\varphi_{v}^{(m+1)}φv(m+1)order m + 1 m + 1 m+1m+1m+1on I. If ∑ v = 0 k a v φ v ( m + 1 ) ( x ) ∑ v = 0 k   a v φ v ( m + 1 ) ( x ) sum_(v=0)^(k)a_(v)varphi_(v)^((m+1))(x)\sum_{v=0}^{k} a_{v} \varphi_{v}^{(m+1)}(x)∑v=0khasvφv(m+1)(x)is non-negative and does not vanish on a dense set in I I III, the function ∑ ν = 0 k a ν φ ν ( x ) ∑ ν = 0 k   a ν φ ν ( x ) sum_(nu=0)^(k)a_(nu)varphi_(nu)(x)\sum_{\nu=0}^{k} a_{\nu} \varphi_{\nu}(x)∑ν=0khasνφν(x)is convex of order m m mmmand so we have inequality ∑ ν = 0 k a ν R [ φ ] > ∑ ν = 0 k   a ν R [ φ ] > sum_(nu=0)^(k)a_(nu)R[varphi] >\sum_{\nu=0}^{k} a_{\nu} R[\varphi]>∑ν=0khasνR[φ]>0.
We leave it to the reader to extend Corollary 1 to this case.
10. As an application, suppose that I I IIIis a positive interval (therefore such that x ∈ I ⇒ x > 0 x ∈ I ⇒ x > 0 x in I=>x > 0x \in I \Rightarrow x>0x∈I⇒x>0) and that S S SSScontains all the power functions x σ x σ x^(sigma)x^{\sigma}xσ, regardless of σ σ sigma\sigmaσreal. All moments (8) are well-defined for σ σ sigma\sigmaσany real number, but relations (9) do not unambiguously determine q v q v q_(v)q_{v}qvthat if ν ν nu\nuνdiffers from integers − 1 , − 2 , … , − m − 1 − 1 , − 2 , … , − m − 1 -1,-2,dots,-m-1-1,-2, \ldots,-m-1−1,−2,…,−m−1Taking this remark into account, it can easily be seen that inequalities (12) are true provided that s s ssseither real but different from − 1 , − 2 … , − m − 1 − 2 r − 1 , − 2 … , − m − 1 − 2 r -1,-2dots,-m-1-2r-1,-2 \ldots,-m-1-2 r−1,−2…,−m−1−2rIn particular, inequality (13) is true if s s sssdiffers from − 1 , − 2 , … , − m − 3 − 1 , − 2 , … , − m − 3 -1,-2,dots,-m-3-1,-2, \ldots,-m-3−1,−2,…,−m−3.
In particular, the linear functional (15) is of the previous form if a 1 , a 2 , … , a n a 1 , a 2 , … , a n a_(1),a_(2),dots,a_(n)a_{1}, a_{2}, \ldots, a_{n}has1,has2,…,hasnare positive numbers.
For example if n = 3 n = 3 n=3n=3n=3And a 1 , a 2 , a 3 a 1 , a 2 , a 3 a_(1),a_(2),a_(3)a_{1}, a_{2}, a_{3}has1,has2,has3are positive and not all equal, we have
(19) q − 1 / 2 2 < q − 3 / 2 q 1 / 2 (19) q − 1 / 2 2 < q − 3 / 2 q 1 / 2 {:(19)q_(-1//2)^(2) < q_(-3//2)q_(1//2):}\begin{equation*} q_{-1 / 2}^{2}<q_{-3 / 2} q_{1 / 2} \tag{19} \end{equation*}(19)q−1/22<q−3/2q1/2
This inequality can easily be reduced to a "basic" inequality. Indeed, we have
q − 3 / 2 = 8 Π ( a 1 + a 2 ) , q − 1 / 2 = 8 3 ⋅ ∑ a 1 a 2 Π ( a 1 + a 2 ) q 1 / 2 = 8 15 ⋅ ∑ a 1 a 2 + 2 ∑ a 1 a 2 a 3 + ∑ a 1 a 1 ( a 2 + a 3 ) Π ( a 1 + a 2 ) q − 3 / 2 = 8 Π a 1 + a 2 , q − 1 / 2 = 8 3 ⋅ ∑ a 1 a 2 Π a 1 + a 2 q 1 / 2 = 8 15 ⋅ ∑ a 1 a 2 + 2 ∑ a 1 a 2 a 3 + ∑ a 1 a 1 a 2 + a 3 Π a 1 + a 2 {:[q_(-3//2)=(8)/(Pi(sqrt(a_(1))+sqrt(a_(2))))","quadq_(-1//2)=(8)/(3)*(sumsqrt(a_(1)a_(2)))/(Pi(sqrt(a_(1))+sqrt(a_(2))))],[q_(1//2)=(8)/(15)*(suma_(1)a_(2)+2suma_(1)sqrt(a_(2)a_(3))+suma_(1)sqrt(a_(1))(sqrt(a_(2))+sqrt(a_(3))))/(Pi(sqrt(a_(1))+sqrt(a_(2))))]:}\begin{aligned} & q_{-3 / 2}=\frac{8}{\Pi\left(\sqrt{a_{1}}+\sqrt{a_{2}}\right)}, \quad q_{-1 / 2}=\frac{8}{3} \cdot \frac{\sum \sqrt{a_{1} a_{2}}}{\Pi\left(\sqrt{a_{1}}+\sqrt{a_{2}}\right)} \\ & q_{1 / 2}=\frac{8}{15} \cdot \frac{\sum a_{1} a_{2}+2 \sum a_{1} \sqrt{a_{2} a_{3}}+\sum a_{1} \sqrt{a_{1}}\left(\sqrt{a_{2}}+\sqrt{a_{3}}\right)}{\Pi\left(\sqrt{a_{1}}+\sqrt{a_{2}}\right)} \end{aligned}q−3/2=8Pi(has1+has2),q−1/2=83⋅∑has1has2Pi(has1+has2)q1/2=815⋅∑has1has2+2∑has1has2has3+∑has1has1(has2+has3)Pi(has1+has2)
where the summons Σ Σ Sigma\SigmaΣand multiplication Π Π Pi\PiPiare extended to circular permutations of indices 1 , 2 , 3 1 , 2 , 3 1,2,31,2,31,2,3.
If we ask x 1 = a 1 , x 2 = a 2 , x 3 = a 3 x 1 = a 1 , x 2 = a 2 , x 3 = a 3 x_(1)=sqrt(a_(1)),x_(2)=sqrt(a_(2)),x_(3)=sqrt(a_(3))x_{1}=\sqrt{a_{1}}, x_{2}=\sqrt{a_{2}}, x_{3}=\sqrt{a_{3}}x1=has1,x2=has2,x3=has3, inequality (19) comes down to inequality (between
(20) ∑ x 1 x 2 3 < ∑ x 1 x 2 ( x 1 2 + x 2 2 ) 6 (20) ∑ x 1 x 2 3 < ∑ x 1 x 2 x 1 2 + x 2 2 6 {:(20)(sumx_(1)x_(2))/(3) < sqrt((sumx_(1)x_(2)(x_(1)^(2)+x_(2)^(2)))/(6)):}\begin{equation*} \frac{\sum x_{1} x_{2}}{3}<\sqrt{\frac{\sum x_{1} x_{2}\left(x_{1}^{2}+x_{2}^{2}\right)}{6}} \tag{20} \end{equation*}(20)∑x1x23<∑x1x2(x12+x22)6
which takes place if the non-negative numbers x 1 , x 2 , x 3 x 1 , x 2 , x 3 x_(1),x_(2),x_(3)x_{1}, x_{2}, x_{3}x1,x2,x3are not all equal.
The direct demonstration of inequality (20) results from inequalities
∑ x 1 x 2 ( x 1 2 + 2 2 ) 6 = 1 3 ∑ x 1 x 2 x 1 2 + x 2 2 2 > 1 3 ∑ x 1 2 x 2 2 > ( 1 3 ∑ x 1 x 2 ) 2 ∑ x 1 x 2 x 1 2 + 2 2 6 = 1 3 ∑ x 1 x 2 x 1 2 + x 2 2 2 > 1 3 ∑ x 1 2 x 2 2 > 1 3 ∑ x 1 x 2 2 (sumx_(1)x_(2)(x_(1)^(2)+_(2)^(2)))/(6)=(1)/(3)sumx_(1)x_(2)(x_(1)^(2)+x_(2)^(2))/(2) > (1)/(3)sumx_(1)^(2)x_(2)^(2) > ((1)/(3)sumx_(1)x_(2))^(2)\frac{\sum x_{1} x_{2}\left(x_{1}{ }^{2}+{ }_{2}{ }^{2}\right)}{6}=\frac{1}{3} \sum x_{1} x_{2} \frac{x_{1}{ }^{2}+x_{2}{ }^{2}}{2}>\frac{1}{3} \sum x_{1}{ }^{2} x_{2}{ }^{2}>\left(\frac{1}{3} \sum x_{1} x_{2}\right)^{2}∑x1x2(x12+22)6=13∑x1x2x12+x222>13∑x12x22>(13∑x1x2)2
Besides q − 3 / 2 , q − 1 / 2 , q − 1 / 2 q − 3 / 2 , q − 1 / 2 , q − 1 / 2 q_(-3//2),q_(-1//2),q_(-1//2)q_{-3 / 2}, q_{-1 / 2}, q_{-1 / 2}q−3/2,q−1/2,q−1/2exist and inequality (19) occurs if a 1 , a 2 , a 3 a 1 , a 2 , a 3 a_(1),a_(2),a_(3)a_{1}, a_{2}, a_{3}has1,has2,has3are non-negative, not all equal, and at most one equal to 0. The preceding results are also valid for the divided difference (15) in general if n > 1 n > 1 n > 1n>1n>1and the a 1 , a 2 , … , a n a 1 , a 2 , … , a n a_(1),a_(2),dots,a_(n)a_{1}, a_{2}, \ldots, a_{n}has1,has2,…,hasnare non-negative, not all equal, and at most one equal to 0. If this last condition is not satisfied, the derivative of order n − 1 n − 1 n-1n-1n−1of the function xa could intervene at point 0 in the reasoning and this derivative may not exist.

BIBLIOGRAPHY

  1. A. Genocchi: Intorno alle funnzioni interpolari. Atti Torino, 13 (1878), 716-730.
  2. GH Hardy, JE Littlewood, G. Pólya: Inequalities. Cambridge, 1934.
  3. T. Popoviciu: On a theorem of Laguerre. Bulletin of the Scientific Society of Cluj (Romania), 8 (1934), 1-4.
  4. T. Popoviciu: On the remainder in certain linear approximation formulas of analysis. Mathematica 1 (24) (1959), 95-142.
  5. T. Popoviciu: Remarks on the remainder of certain formulas for approximating a difference divided by derivatives. Buletinul Institutului Politehnic din Iaşi 8 (17), Fasc. 3-4 (1967), 103-109.
  6. T. Popoviciu: On the form of the remainder of certain quadrature formulas. Proccedings of the Conference on Constructive Theory of Functions, (1969), 365-370.
Calculation Institute
CLUJ
Romania
    • Presented on September 1, 1972 by DS Mitrinović.
  2. *) My work was received by the editors of the journal where it was published on January 31, 1934. The preface to the book [2] by HLP is dated July 1934.
1972

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