Characterization of homogeneous symmetric monotone bivariate means
Raïssouli and Rezgui Journal of Inequalities and Applications
Characterization of homogeneous symmetric monotone bivariate means
Mustapha Raïssouli 1 2
Anis Rezgui 0 2
0 Mathematics Department, INSAT, Carthage University , Tunis , Tunisia
1 Department of Mathematics, Science Faculty, Moulay Ismail University , Meknes , Morocco
2 Department of Mathematics, Science Faculty, Taibah University , P.O. Box 30097, Al Madinah Al Munawwarah, 41477 , Saudi Arabia
In this paper, we introduce a class of bivariate means generated by an integral of a continuous increasing function on (0, +∞). This class of means widens the spectrum of possible means and leads to many easy and interesting mean-inequalities. We show that this class of means characterizes the large class of homogeneous symmetric monotone means.
bivariate mean; differential equation; intrinsic function
-
(.)
A mean is said to be strict if the inequalities in (.) are both strict for all a = b. A
continuous (resp. symmetric/homogeneous) mean is defined as usual; see, for instance, [].
Standard examples of such means are the following:
A := A(a, b) = a + b ;
S := S(a, b) = √a + √b
;
Q := Q(a, b) =
√
G := G(a, b) = ab;
H := H(a, b) = a+abb ;
b – a
L := L(a, b) = ln b – ln a ,
L(a, a) = a;
a + b
;
a + b
C := C(a, b) = a + b ,
which are known in the literature as the arithmetic mean, geometric mean, harmonic
mean, square-root mean, logarithmic mean, quadratic (or root-square) mean, and
contraharmonic mean, respectively. For more examples and details about bivariate means and
their applications, we refer the reader to [] and the references therein.
An interesting example of nonsymmetric homogeneous mean is the so-called
SchwabBorchardt mean, denoted by SB, and defined as [, ]
⎧ √b–a
SB := SB(a, b) = ⎨ cos–(a/b)
√a–b
⎩ cosh–(a/b)
with SB(a, a) = a. This nonsymmetric mean stems its importance in the fact that it includes
a lot of symmetric means in the sense that
L = SB(A, G),
P = SB(G, A),
T = SB(A, Q),
M = SB(Q, A),
where
P := P(a, b) = tan–a(√–ab/b) – π =
a – b a – b
sin–( aa–+bb ) = tan–( √√aa–+√√bb ) ,
T := T (a, b) =
M := M(a, b) =
taan––(baa–+bb ) = tan–a(a–/bb) – π / =
a – b
sin–( aa–+bb ) ,
a – b
sinh–( aa–+bb ) ,
M(a, a) = a;
P(a, a) = a;
T (a, a) = a;
they are, respectively, known as the first Seiffert mean [], the second Seiffert mean [],
and the Neuman-Sándor mean []. For more details about recent developments for SB,
see, for instance, [, , ]. The previous standard means satisfy the well-known chain of
inequalities
H < G < L < P < A < M < T < Q,
where the notation m < m, between two means m and m, means that m(a, b) < m(a, b)
for all a, b > with a = b.
For a homogeneous mean m, we define its associate function φm : (, ∞) −→ (, ∞) via
the relationship m(a, b) = bm(a/b, ) = bφm(a/b), that is, φm(x) = m(x, ) for all x > . In this
case, (.) yields
x ≤ φm(x) ≤ if < x ≤ ,
≤ φm(x) ≤ x if x ≥ .
(.)
(.)
(.)
A homogeneous mean m is called monotone if φm is increasing on (, ∞). In what follows,
if there is no confusion, then we write φ instead of φm. The means A, G, H, S, Q, and L are
monotone, whereas C is not; see [] for more details.
2 A new class of bivariate means
As mentioned before, in the sequel, we introduce and investigate a new class of means
generated by an integral of a function. Let f be a continuous strictly monotonic function
on the open interval (, ∞). For every a, b > , we define
Proposition . For any continuous and strictly monotonic function f on (, +∞), the
binary map mf defined by (.) is a homogeneous bivariate mean.
Proof It is straightforward and therefore omitted here.
Remark . The mean mf is not always symmetric; see Example . and Example ..
Otherwise, it is not hard to check that mf +c = mf for each constant c and that mα,f =
mf for every α = . In particular, m–f = mf . Due to this, without loss of generality, we
only consider functions f that are continuous strictly increasing on (, ∞) and satisfying
f () = .
In the sequel, we will use the following notations:
C↑(, ∞) = f : (, ∞) −→ f (, ∞) : f is continuous and strictly increasing
and
C↑(, ∞) = f ∈ C↑(, ∞) : f is continuously differentiable .
The following result gives other equivalent forms of mf .
Proposition . Let f ∈ C↑(, ∞) be such that f () = . Then the following assertions
hold:
(i) For all a, b > , we have
mf (a, b) = b
f – tf (a/b) dt.
Proof (i) If in (.) we make the change of variables u = tf (a/b) with ≤ t ≤ , then we
obtain (.) by simple topics of integration.
(ii) Setting u = f (s) in (.), we obtain (.) after an elementary manipulation. By
integration by parts, (.) follows from (.).
The previous forms of mf lead to the following regularity result.
Corollary . Let f ∈ C↑(, ∞) be such that f () = . Then the mean mf is continuous
strict monotone.
Proof Since f is continuous and mf is a mean, (...truncated)