Characterization of homogeneous symmetric monotone bivariate means

Journal of Inequalities and Applications, Sep 2016

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. MSC: 26E60.

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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)


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Mustapha Raïssouli, Anis Rezgui. Characterization of homogeneous symmetric monotone bivariate means, Journal of Inequalities and Applications, 2016, pp. 217, 2016, DOI: 10.1186/s13660-016-1150-9