1959 a -Popoviciu- Mathematica - Some inequalities between means (in Russian)
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CONCERNING SOME INEQUALITIES BETWEEN MEANS
TIBERIU POPOVIC
Cluj
Let a sequence (finite or infinite) of numbers
(1) be given.a_(1),a_(2),dotsa_{1}, a_{2}, \ldots
Assuming that the terms of sequence (1) are non-negative, letA_(n)=(a_(1)+a_(2)+dots+a_(n))/(n)A_{n}=\frac{a_{1}+a_{2}+\ldots+a_{n}}{n}arithmetic mean andG_(n)=root(n)(a_(1)a_(2)dotsa_(n))G_{n}=\sqrt[n]{a_{1} a_{2} \ldots a_{n}}geometric mean of the firstnnmembersa_(1),a_(2),dots,a_(n)a_{1}, a_{2}, \ldots, a_{n}of this sequence. L. Chakalov proved [1] the following two properties:
I. Sequence
(2)n(A_(n)-G_(n)),quad n=1,2,dotsn\left(A_{n}-G_{n}\right), \quad n=1,2, \ldots
is non-decreasing.
II. If sequence (1) is non-decreasing, then sequence
(3)(n^(2))/(n-1)(A_(n)-G_(n)),quad n=2,3,dots\frac{n^{2}}{n-1}\left(A_{n}-G_{n}\right), \quad n=2,3, \ldots
is also non-decreasing.
To generalize these properties, we studied the monotonicity of sequences obtained from (2), (3), replaced the geometric meanG_(n)G_{n}through "quasi-arithmetic", more general.
Throughout we assume thatf=f(x)f=f(x)is a function of one real variablexxand continuous on the open intervalII. Ifffstrictly monotone (increasing or decreasing) function, then its inverse functionF=F(x)F=F(x)is also defined, continuous and strictly monotone in the same sense asff(i.e. increasing respectively decreasing) on the open intervalI^(')I^{\prime}In particular, this is the case ifffcloud-
gives a derivative that does not vanish onIIIn this caseFFalso has a derivative that does not vanish over the entire intervalI^(')I^{\prime}, and has the same sign as the derivative functionff.
If the members of the sequence (1) belong to the intervalII, and ifffstrictly monotone function, then the quasi-arithmetic mean
firstnnmembers of the sequence (1) are defined forn=1,2,dotsn=1,2, \ldots
Before going further, we note that in some cases we can continue the definition of a function by continuityffand averageM_(n)(f)M_{n}(f)at one end of the intervalII, assumed to be finite. In this case, the functionffand the average (4) will have a very definite value (either proper or improper), even if some or all numbersaacoincide with such an end. For example, in the case of the geometric mean, to which the mean (4) for the function is reducedf=ln xf=\ln xwe can takeM_(n)(f)=-0M_{n}(f)=-0, if at least one of the numbersa_(1),a_(2),dots,a_(n)a_{1}, a_{2}, \ldots, a_{n}is equal to 0 . Below we will not dwell on such continuations, because the results related to the monotonicity of sequences of the type under consideration retain, generally speaking, their force and are obtained by passing to the limit. Thus, for example, it is sufficient to prove the monotonicity of sequence (2) in the case of positivity of alla_(i)a_{i}and from this we conclude that the property remains true even in the case when the numbersaanon-negative.
3. - Has a place following
THEOREM 1. - If a functionffstrictly monotone if the term of the sequence (1) belongs to the interval I and if the sequence (1) is monotone, then the sequence
also monotone in the same sense.
If, under these conditions, the members of sequence (1) are not all equal between the messages, then sequence (5) is strictly monotone, starting from some indexn^(**)n^{*}).
The conditions of the theorem distinguish 4 alternatives, since the functionffmay be increasing or decreasing, and the sequence (1) may be non-decreasing or increasing.
Let us prove the theorem under the assumption thatffan increasing function and the sequence (1) is non-decreasing. Let, in general,
Ifn >= mn \geq m, then equality is impossible for any valueffin formulas (6), and therefore it is not possible either in formula (7) or in formula (8).
Thus, the theorem is proven in the alternative under consideration. The theorem is proved similarly in the other three alternatives.
4. - Below, we will need some well-known properties of ordinary convex functions (first order) or second-order convex functions.
Functionvarphi=varphi(x)\varphi=\varphi(x), defined on the intervalII, is called non-concave, respectively non-convex of the same order, if its divided difference(n+1)(n+1)-th order on any points (different) fromIIremains non-negative, respectively, non-positive, all the time. If we denote by[x_(1),x_(2),dots,x_(n+2);varphi]\left[x_{1}, x_{2}, \ldots, x_{n+2} ; \varphi\right]divided difference(n+1)(n+1)-th order functionvarphi\varphiat points or nodes (various)x_(1),x_(2),dots,x_(n+2)x_{1}, x_{2}, \ldots, x_{n+2}, then non-concavity, respectively non-convexitynn-th order functionvarphi\varphion the intervalIIcharacterized by inequality
at any (different) nodesx_(1),x_(2),dots,x_(n+2)in Ix_{1}, x_{2}, \ldots, x_{n+2} \in I.
In particular, if the equal sign in (9) never occurs, then the functionvarphi\varphiis called convex, respectively concavenn-th order onII.
Ifn=0n=0then we have a non-decreasing or non-increasing function, and, in particular, an increasing, respectively, decreasing function. Thus, functions of zero order are monotone functions.
In order for it to be continuous on the intervalIIfunctionvarphi\varphiwas non-concave, respectively non-convexnn-th order, it is necessary and sufficient that inequality (9) holds only at equidistant nodes. More precisely, formula
(10)
Where
[x,x+h,dots,x+ bar(n+1)h;varphi]=(1)/((n+1)!h^(n+1))Delta_(h)^(n+1)varphi(x),[x, x+h, \ldots, x+\overline{n+1} h ; \varphi]=\frac{1}{(n+1)!h^{n+1}} \Delta_{h}^{n+1} \varphi(x),
shows that in order for continuous onIIfunctionvarphi\varphiwas convex, non-concave, non-convex, respectively concavenn-th order onIIit is necessary and sufficient to have
under any{:x,x+ bar(n+1)h in I,h > 0^(**))\left.x, x+\overline{n+1} h \in I, h>0^{*}\right).
*) If oddnnconditionh > 0h>0can be replaced by a conditionalh!=0h \neq 0.
Any given onllorder functionn > 0n>0is continuous onI^(**)I^{*}). Any givenIIorder functionn > 1n>1admits a continuous derivative onII. If the functionvarphi\varphiconvex, non-concave, non-convex, respectively concave ordern >= 1n \geqq 1, then its derivativevarphi^(')\varphi^{\prime}provided that it exists, is convex, non-concave, non-convex, respectively concave of ordern-1n-1. Inequalityp^((n+1)) >= 0p^{(n+1)} \geqq 0resp.<= 0\leqq 0for anyonex epsilon Ix \epsilon Inecessary and sufficient for non-concavity, respectively non-convexity of ordernnfunctionvarphi\varphionII. Inequalityvarphi^((n+1)) > 0\varphi^{(n+1)}>0resp.< 0<0, for anyonex in Ix \in I, it is sufficient for the convexity respectively concavity of the functionvarphi\varphionII.
5. - Let us return to L. Chakalov's property. A generalization of the sequence (2) is the sequence
Then the following THEOREM 2 holds.
- If the functionffstrictly monotone, then in order for the sequence (11) to be non-decreasing, respectively, non-increasing for any sequence (1) whose members belong to I, it is necessary and sufficient that the function be non-concave, respectively, non-convex of order 1FFinverse to the functionff.
and throughD(a_(1),a_(2),dots,a_(n))=n(A_(n)-M_(n)(f))-(n-1)(A_(n-1)-M_(n-1)(f))D\left(a_{1}, a_{2}, \ldots, a_{n}\right)=n\left(A_{n}-M_{n}(f)\right)-(n-1)\left(A_{n-1}-M_{n-1}(f)\right)the difference between two consecutive terms of the sequence (11). We have
we deduce that the numbersB_(n-1),B_(n),f(a_(n))B_{n-1}, B_{n}, f\left(a_{n}\right)are different or all equal to each other depending on whether they are equalf(a_(n))=B_(n-1)f\left(a_{n}\right)=B_{n-1}or not. Therefore,
where the right side of the equality is replaced by 0 iff(a_(n))=B_(n-1)f\left(a_{n}\right)=B_{n-1}, from which the sufficiency of the condition of the theorem follows.
Ifx,x+2h in Ix, x+2 h \in I, and if we takea_(1)=F(x),a_(2)=F(x+2h)a_{1}=F(x), a_{2}=F(x+2 h)then we haveD(a_(1),a_(2))=Delta_(h)^(2)F(x)D\left(a_{1}, a_{2}\right)=\Delta_{h}^{2} F(x)from this it immediately follows that the condition of the theorem is necessary. Theorem 2 is proved.
6. - To generalize L. Chakalov's property, instead of sequence (3), we take the sequence
The following THEOREM 3 holds
. - The functionFFstrictly monotone onI^(')I^{\prime}, then for the sequence (14) to be non-decreasing for any non-decreasing sequence (1) whose terms belong to I, it is sufficient that the inverse function F be:
a) non-concave of order1u1 u
b) non-concave, respectively non-convex of order 2 depending on whetherffincreasing or decreasing function.
For proof we will use the method given by L. Chakalov for sequence (3).
Let us first assume thatFF, therefore andff, has an arbitrary non-vanishing onI^(')I^{\prime}, respectively onII.
the difference between two consecutive terms of the sequence (14).
For everyonek=0,1,dots,n-1k=0,1, \ldots, n-1functionE_(k)=E_(k)(x)E_{k}=E_{k}(x), resulting fromE(a_(1),a_(2),dots,a_(n))E\left(a_{1}, a_{2}, \ldots, a_{n}\right)believing in the lattera_(k+1)=a_(k+2)=dots=a_(n)=xa_{k+1}=a_{k+2}=\ldots=a_{n}=xcontinuous and differentiable onII.
Below it is enough to assumea_(1) <= a_(2) <= dots <= a_(n)a_{1} \leqq a_{2} \leqq \ldots \leqq a_{n}Let
's consider the average (12), in which it is assumeda_(k+1)=a_(k+2)=dots==a_(n)=xa_{k+1}=a_{k+2}=\ldots= =a_{n}=x. Becausef^(')(x)F^(')(f(x))=1f^{\prime}(x) F^{\prime}(f(x))=1a simple calculation gives the derivative of the functionE_(k)(x)E_{k}(x)in the form
show that whenkappa\kappapoints>= 1\geqq 1Andx > a_(k),f(x),B_(n),B_(n-1)x>a_{k}, f(x), B_{n}, B_{n-1}fromI^(')I^{\prime}are different becausef(x) > B_(k)f(x)>B_{k}resp.f(x) < B_(k)f(x)<B_{k}depending on whether ffincreasing or decreasing. A simple calculation, which we will not reproduce here, gives
Butf^(')(x),f(x)-B_(k)f^{\prime}(x), f(x)-B_{k}positive, respectively negative, depending on whether it isff, henceFF, increasing or decreasing. Formula (15) shows that ifFFsatisfies conditions a) and b) of Theorem 3, then we haveE_(k)^(') >= 0E_{k}^{\prime} \geqq 0Forx > a_(k)x>a_{k}Andk=1,2,dots,n-1k=1,2, \ldots, n-1, It follows that the functionsE_(k),k=1,2,dots,n-1E_{k}, k=1,2, \ldots, n-1non-decreasing forx >= a_(k)x \geqq a_{k}From this we conclude
>= E(a_(1),a_(2),dots,a_(n-3),a_(n-2),a_(n-2),a_(n-2)) >= dots >= E(a_(1),a_(1),dots,a_(1))=E_(0)(a_(1))\geqq E\left(a_{1}, a_{2}, \ldots, a_{n-3}, a_{n-2}, a_{n-2}, a_{n-2}\right) \geqq \ldots \geqq E\left(a_{1}, a_{1}, \ldots, a_{1}\right)=E_{0}\left(a_{1}\right)
ButE_(0)=0E_{0}=0, whatever it may bexx, hence,E(a_(1),a_(2),dots,a_(n)) >= ,0E\left(a_{1}, a_{2}, \ldots, a_{n}\right) \geqq, 0, which is what was required to be proven.
It is easy to see that ifFFMoreover, the convex sequence (1) of order 2 and non-decreasing has all terms equal to each other, or ifFFconvex of order 1 and non-decreasing sequence (1) has at least three numerically distinct terms, then sequence (14) is increasing, starting from some indexnn.
Now we can remove the restriction made at the beginning of the proof. The functionFF, being of order 2, is continuous onI^(')I^{\prime}For the sake of certainty, let us assume thatFFincreasing function. Then the functionF_(epsi)=F+epsi xF_{\varepsilon}=F+\varepsilon xincreasing and satisfies conditions a), b) of the Gris theorem in anyepsi > 0\varepsilon>0. In addition, the derivative functionF_(epsi)F_{\varepsilon}does not vanish onI^(')I^{\prime}. Therefore, we can apply the theorem to the functionF_(epsi)F_{\varepsilon}But whenepsi rarr0\varepsilon \rightarrow 0, functionF_(epsi)F_{\varepsilon}uniformly tends toFFon any finite and closed interval. Iff_(epsi)f_{\varepsilon}there is an inverse function to the functionF_(epsi)F_{\varepsilon}, Thenf_(epsi)f_{\varepsilon}uniformly tends toffon the same interval. From this it follows directly thatM_(n)(f_(epsi))M_{n}\left(f_{\varepsilon}\right)strives forM_(n)(f)M_{n}(f)atepsi rarr0\varepsilon \rightarrow 0The reasoning is similar ifFFdecreasing function, but under the assumption thatepsi < 0\varepsilon<0. Thus, Theorem 3 can be obtained by passing to the limit from the already proven special case.
7. - From sequences (1) and (14) we can require non-decreasing or non-increasing, therefore, the satisfaction of one of the 4 alternatives which we will denote bym_(1),m_(2),m_(3)m_{1}, m_{2}, m_{3}Andm_(4)m_{4}and which we present in the table.
Theorem 3 refers to the alternativem_(1)m_{1}A similar theorem can be proved for each of the other alternatives. In the case of alternativem_(2)m_{2}conditions a) and b) that the function satisfiesFF, are replaced by the following conditions:
a')FFnon-convex of order 1.
b')FFnon-concave, respectively non-convex of order 2, depending on whether it is an increasing or decreasing function.
In alternativesm_(3)m_{3}Andm_(4)m_{4}the sign of the derivatives (15) is studiedx < a_(k)x<a_{k}, where we now assumea_(1) >= a_(2) >= dots >= a_(n)a_{1} \geqq a_{2} \geqq \ldots \geqq a_{n}In these casesf^(')(x),f(x)cdotsB_(k)f^{\prime}(x), f(x) \cdots B_{k}have different signs forx < a_(k)x<a_{k}. Therefore, conditions a) and b), which the function satisfies, must be replaced by conditions a) and b') for the alternativem_(3)m_{3}, and conditions a') and b) of the alternativem_(4)m_{4}.
8. - The question arises whether conditions a) and b) of Theorem 3 are also necessary for the non-decreasing nature of sequence (14), when sequence (1) is non-decreasing. For sequence (14) to be non-decreasing, it is necessary, in particular, to have
for anyx_(1) <= x_(2) <= x_(3)x_{1} \leqq x_{2} \leqq x_{3}, Ifffincreasing, and for anyx_(1) >= x_(2) >= x_(3)x_{1} \geqq x_{2} \geqq x_{3}, есіи{\{decreasing.
Ifffincreasing, putting firstx_(1)=x_(2)=x,x_(3)=x+3hx_{1}=x_{2}=x, x_{3}=x+3 hand thenx_(1)=x,x_(2)=x_(3)=x+6hx_{1}=x, x_{2}=x_{3}=x+6 hwe deduce that the functionFFmust satisfy the inequalities.
for anyx,hx, hsuch thath > 0h>0Andx,x+3h inI^(')x, x+3 h \in I^{\prime}, respectively.x,x+6h inI^(')x, x+6 h \in I^{\prime}This result is obtained from the fact that, based on formula (10), the left-hand sides of inequalities (16), (17) can be replaced by
In the same way, ifffdecreasing, assuming firstx_(1)=x_(2)x_{1}=x_{2}and thenx_(2)=x_(3)x_{2}=x_{3}, we deduce that the functionFFmust satisfy the inequalities
{:[(18)[x","x+2hx+3h;F] >= 0],[(19)[x","x+2h","x+3h","x+6h;F] <= 0]:}\begin{gather*}
{[x, x+2 h x+3 h ; F] \geqq 0} \tag{18}\\
{[x, x+2 h, x+3 h, x+6 h ; F] \leqq 0} \tag{19}
\end{gather*}
under the same conditions relative toxxAndhh.
9. - It takes place
LEMMA 1. - If c is continuous onff, then a necessary and sufficient condition for convexity, non-concavity, non-convexity, respectively concavity of order 1 of a function o on I consists in the fulfillment of the inequalities
under anyx,x+3h in I,h > 0x, x+3 h \in I, h>0.
Obviously, the condition is necessary. Let us prove its sufficiency. Note that convexity, respectively, concavity, are special cases of non-concavity, respectively, non-convexity (of the same meaning, that by changing the sign of the functionvarphi\varphi(taking the function -varphi\varphiinstead of a functionvarphi\varphi) the meaning of its convexity is replaced: convexity by concavity and, in general, non-concave by non-convexity. Further, if the functionvarphi\varphihas order 1, and[x_(1),x_(2),x_(3);varphi]=0\left[x_{1}, x_{2}, x_{3} ; \varphi\right]=0for these pointsx_(1) < x_(2) < x_(3)x_{1}<x_{2}<x_{3}then we have[x_(1)^('),x_(2)^('),x_(3)^(');varphi]=0\left[x_{1}^{\prime}, x_{2}^{\prime}, x_{3}^{\prime} ; \varphi\right]=0at any different pointsx_(1)^('),x_(2)^('),x_(3)^(')in[x_(1),x_(3)]x_{1}^{\prime}, x_{2}^{\prime}, x_{3}^{\prime} \in\left[x_{1}, x_{3}\right]. Therefore, it is enough to prove that if
{:(20)[x","x+h","x+3h;varphi] >= 0","quad x","x+3h in I","h > 0:}\begin{equation*}
[x, x+h, x+3 h ; \varphi] \geqq 0, \quad x, x+3 h \in I, h>0 \tag{20}
\end{equation*}
then the functionvarphi\varphinon-concave of order 1.
Ifx_(1) < x_(2) < dots < x_(m+2),(m >= 1)x_{1}<x_{2}<\ldots<x_{m+2},(m \geqq 1), we have the following formula for the average
gamma_(i)=((x_(i+1)-x_(1))(x_(i+2)-r_(i)))/((x_(m+1)-x_(1))(x_(m+2)-x_(1))),i=1,2,dots,m\gamma_{i}=\frac{\left(x_{i+1}-x_{1}\right)\left(x_{i+2}-r_{i}\right)}{\left(x_{m+1}-x_{1}\right)\left(x_{m+2}-x_{1}\right)}, i=1,2, \ldots, m
Taking into account the continuity of the functionvarphi\varphi, we can go to the limit atm rarr oom \rightarrow \infty, and get[x,x+h,x+2h;varphi] >= 0[x, x+h, x+2 h ; \varphi] \geqq 0. Hence,Delta_(h)^(2)varphi(x) >= 0\Delta_{h}^{2} \varphi(x) \geqq 0Forx,x+2h epsilon Ix, x+2 h \epsilon I, from which, based on what was said in No. 4, follows the non-concavity of order 1 of the functionvarphi\varphiThus, Lemma 1 is proved.
Now we can deduce
Corollary 1. - Condition a) imposed on the functionFFfrom Theorem 3, it is necessary for the sequence (1.4) to be non-decreasing for any non-decreasing sequence (1).
Ifffincreasing function, then the required property follows from inequality (16) and Lemma I applied to the functionFF. Ifffnon-increasing, then the functionF(x),F(-x)F(x), F(-x)are simultaneously non-concave or non-convex of order 1, in addition, if the functionF(x)F(x)satisfies inequality (18), ts functionF(-x)F(-x)satisfies inequality (16). Corollary 1 is also obtained in this case from Lemma 1. One can proceed either by repeating the entire argument forh < 0h<0instead ofh > 0h>0by preno nor o, tivo directly establishing a similar lemma for divided differences of the form
[x,x+2h,x+3h;varphi]" с "h > 0.[x, x+2 h, x+3 h ; \varphi] \text { с } h>0 .
It is similarly proved that conditions a), a') in the corresponding theorems relating to alternativesm_(2),m_(3)m_{2}, m_{3}Andm_(4)m_{4}, such are necessary.
11. - We will prove only for one special case that condition b) from Theorem 3 is also necessary. Im
Corollary 2. - Condition b), prescribed for the functionFFin Theorem 3, whereFFhas a continuous third-order derivative onI^(')I^{\prime}, is necessary for the non-decreasing sequence (14) for any non-decreasing sequence (1).
Suppose f is an increasing function. We will show that the derivativeF^(''')F^{\prime \prime \prime}, assuming that it exists and is continuous onI^(')I^{\prime}, is non-negative onI^(')I^{\prime}Let's assume the opposite. Then there is a subinterval[alpha,beta][\alpha, \beta]intervalI^(')I^{\prime}, Withalpha < beta\alpha<\betasuch that we have on itF^(''') < 0F^{\prime \prime \prime}<0. But then the functionFFconcave of order 2 on[alpha,beta][\alpha, \beta], therefore ifxx,x+6h in[alpha,beta],h > 0x+6 h \in[\alpha, \beta], h>0, we have[x,x+3h,x+4h,x+6h;F] < 0[x, x+3 h, x+4 h, x+6 h ; F]<0which contradicts inequality (17). So,F^(''') >= 0F^{\prime \prime \prime} \geq 0Forx inI^(')x \in I^{\prime}It is concluded thatFFnon-concave of order 2 onI^(')I^{\prime}, thus Corollary 2 is proven.
In the same way, relying on inequality (19), one can prove Corollary 2, whenffdecreasing function.
It is similarly proved that conditions b), b') of the theorems corresponding to the alternativesm_(2),m_(3)m_{2}, m_{3}Andm_(4)m_{4}are also necessary under the assumption thatFFhas a continuous third order.
12. - Using the introduced notations, let us consider, instead of the sequence (11), the succession
This sequence is reduced to (11) if sequence (1) is replaced by sequence (13) and if the functions are interchangedffAndFF. From Theorem 2, therefore, follows a similar property relating to sequence (23). However, a somewhat more general property holds, since sequence (23) is simpler in some sense than sequence (11), which does not contain the inverse function toff. It can be foreseen that no character of monotonicity of the functionffhas no obligatory influence.
The following THEOREM 4 holds
. - In order for sequence (23) to be non-decreasing, respectively, non-increasing for any sequence (1) whose terms belong to the interval 1, it is necessary and sufficient that the functions f be non-concave, respectively, non-convex of order 1.
As in the case of Theorem 2, the proof will be immediate if we note that the difference between two consecutive terms (betweennn-ym and (n-1n-1)-th) of sequence (23) is equal to
which is reduced to (14) using the same transformations that reduced sequence (23) to (11).
There is a property similar to that contained in Theorem 3. Namely,
THEOREM 5 - In order for sequence (24) to be non-decreasing for any non-decreasing sequence (1) whose terms belong to the interval I, it is sufficient that the function f be
a) non-concave of order 1 and
c) non-concave of order 2.
The proof is the same as in Theorem 3. Therefore, it is sufficient to indicate only its idea.
Let us denote byPhi(a_(1),a_(2),dots,a_(n))\Phi\left(a_{1}, a_{2}, \ldots, a_{n}\right)the difference between two consecutive terms (between (n-1n-1)-th and (n-2n-2)-th,n > 2n>2) sequence (24). LetPhi_(k)=Phi_(k)(x)\Phi_{k}=\Phi_{k}(x)function ofxx, which is plouchasya kzPhi(a_(1),a_(2),dots,a_(n))\Phi\left(a_{1}, a_{2}, \ldots, a_{n}\right)substitutiona_(k+1)=a_(k+2)=dots=a_(n)=xa_{k+1}=a_{k+2}=\ldots=a_{n}=xFinally, to prove it, let us assume thata_(1) <= a_(2) <= dots <= a_(n)a_{1} \leqq a_{2} \leqq \ldots \leqq a_{n}.
Functionff, which has order 2, has a continuous derivative onII. It follows that the functionsPhi_(k)\Phi_{k}are continuous and have a continuous derivative onII.
Ifk >= 1,a_(k) < x in Ik \geqq 1, a_{k}<x \in I, then the pointsx,A_(n),A_(n-1)x, A_{n}, A_{n-1}fromIIthe derivatives of the function are different in the casephi_(k)\phi_{k}in the form
it follows that these derivatives are non-negative whenx > a_(k)x>a_{k}. Like Theorem 3, No. 6, Theorem 5 is obtained from the corresponding formulas.
It is easy to notice here that if the functionffin addition, it is also convex of order 2 and the non-decreasing sequence (1) does not have all terms equal to each other, or if the functionint\intis convex of order 1 and the non-decreasing sequence (1) has at least three numerically distinct terms, then the sequence (24) will be increasing, starting from some indexnn.
14. - Here, as in No. 7, there are four alternatives.m_(1),m_(2)m_{1}, m_{2},m_(3)m_(4)m_{3} m_{4}we only need to replace sequence (14) with sequence (24). Theorem 5 corresponds to the alternativem_(1)m_{1}In a similar theorem, corresponding to the alternativem_(2)m_{2}, conditions a), c), which the function satisfiesffin Theorem 5, are replaced by the following conditions: {:a^('))\left.\mathrm{a}^{\prime}\right)f is non-convex of order 1,
c')ffis non-convex of order 2.
In similar theorems corresponding to alternativesm_(3)m_{3}Andm_(4)m_{4}, conditions a), c) of Theorem 5 are replaced by a), b, ') and a'), c) respectively.
Finally, as in No. 10, it is proved that condition a) of Theorem 5 is necessary for the non-decreasing nature of sequence (24) for any non-decreasing sequence (1). As in No. 11, it is proved that for the same result, condition c) is necessary ifffthree times continuously differentiable function onII.
Similar properties are preserved in the case of alternativesm_(2),m_(3),m_(4)m_{2}, m_{3}, m_{4}.
15. - Properties I and II of L. Chakalov are derived from Theorems 2 and 3 atf=ln xf=\ln x. In this case, the functionffis atx > 0x>0increasing, concave of order 1 and convex of order 2. The inverse functionF=e^(x)F=e^{x}increasing, convex of order 1 and convex of order c. We can now apply Theorems 4 and 5, noting that for the latter we are in the alternativem_(4)m_{4}. Preliminarily moving from logarithms to numbers, we thus obtain the following two properties:
III. - Sequence((A_(n))/(G_(n)))^(n),n=1,2,dots\left(\frac{A_{n}}{G_{n}}\right)^{n}, n=1,2, \ldots- non-decreasing.
IV. - Sequence((A_(n))/(G_(n)))^((n^(a))/(n-1)),n=2,3,dots\left(\frac{A_{n}}{G_{n}}\right)^{\frac{n^{a}}{n-1}}, n=2,3, \ldots- non-decreasing for any non-decreasing sequence (1).
Here we have used the notation kz № 1 for the arithmetic mean and geometric mean, assuming that all members of the sequence (1) are positive.
Let's take another case, namelyf=(1)/(x)f=\frac{1}{x}atx > 0x>0We have thenF=(1)/(x)F=\frac{1}{x}Quasi-arithmetic meanM_(n)(f)M_{n}(f)then reduces to the harmonic meanH_(n)=(n)/((1)/(a_(1))+(1)/(a_(2))+dots+(1)/(a_(n)))H_{n}=\frac{n}{\frac{1}{a_{1}}+\frac{1}{a_{2}}+\ldots+\frac{1}{a_{n}}}firstnnmembers of the sequence (1), assumed to be positive.
Both functionsf,Ff, Fare decreasing, convex of order 1 and concave of order 2. Now we can apply the theorems2,3,42,3,4and 5, noting that for Theorem 3 we are in the alternativem_(1)m_{1}, and for Theorem 5 - in the alternativem_(3)m_{3}. We thus derive the properties:
V. - Sequences
are non-decreasing, the first for any non-decreasing sequence (1).
Finally, let's consider one more special case:f=x^(2)f=x^{2}, atx > 0x>0We have thenF=sqrtxF=\sqrt{x}and averageM_(n)(f)M_{n}(f)is reduced to a quadratic meanP_(n)=sqrt((a_(1)^(2)+a_(2)^(2)+dots+a_(n)^(2))/(n))P_{n}=\sqrt{\frac{a_{1}^{2}+a_{2}^{2}+\ldots+a_{n}^{2}}{n}}firstnnnumbers of the sequence (1).
Here is the functionffincreasing, convex of order 1 and at the same time non-concave and non-convex row 2, and the functionFF- increasing, concave of order 1 and convex of order 2.
We can apply Theorems 2 and 3; in the latter case we are in the alternativem_(4)m_{4}. We thus obtain the properties:
VII. - Sequencen(A_(n)-P_(n)),n=1,2,dotsn\left(A_{n}-P_{n}\right), n=1,2, \ldotsis non-increasing.
VIII. - Sequence(n^(2))/(n-1)(A_(n)-P_(n)),n=1,2,dots\frac{n^{2}}{n-1}\left(A_{n}-P_{n}\right), n=1,2, \ldotsis non-increasing for any non-increasing sequence (1).
We can also apply Theorems 4 and 5, noting that in the latter case we are in the alternativesm_(1)m_{1}Andm_(3)m_{3}. We then obtain the properties:
IX. - Sequencen(P_(n)^(2)-A_(n)^(2)),n=1,2,dotsn\left(P_{n}^{2}-A_{n}^{2}\right), n=1,2, \ldotsis non-decreasing.
X. - Sequence(n^(2))/(n-1)(P_(n)^(2)-A_(n)^(2)),n=2,3,dots\frac{n^{2}}{n-1}\left(P_{n}^{2}-A_{n}^{2}\right), n=2,3, \ldotsis non-decreasing for any monotone sequence (1).
In properties VII-X, we can assume that the terms of sequence (1) are non-negative.
Properties IX and X can be obtained directly and very simply if we note that
