Optimal Bounds for Neuman Mean Using Arithmetic and Centroidal Means

Abstract

We present the best possible parametersα1,α2,β1,β2∈Randα3,β3∈(1/2,1)such that the double inequalitiesα1A(a,b)+(1-α1)C(a,b)<NQA(a,b)<β1A(a,b)+(1-β1)C(a,b),Aα2(a,b)C1-α2(a,b)<NQA(a,b)<Aβ2(a,b)C1-β2(a,b),andC[α3a+(1-α3)b,α3b+(1-α3)a]<NQA(a,b)<C[β3a+(1-β3)b,β3b+(1-β3)a]hold for alla,b>0witha≠band give several sharp inequalities involving the hyperbolic and inverse hyperbolic functions. Here,N(a,b),A(a,b),Q(a,b), andC(a,b)are, respectively, the Neuman, arithmetic, quadratic, and centroidal means ofaandb, andNQA(a,b)=N[Q(a,b),A(a,b)].

Optimal bounds of exponential type for arithmetic mean by Seiffert-like mean and centroidal mean

“…For example, two common means can be used to define some new means. The recent success in this respect can be seen in References [5][6][7][8][9][10] . In [1] Witkowski introduced the following two new means, one called sine mean Recently, Nowicka and Witkowski [2] determined various optimal bounds for the M sin (x, y) and M tanh (x, y) by the arithmetic mean A(x, y) = (x + y)/2 and centroidal mean C e (x, y) = (2/3) (x 2 + x y + y 2 )/ (x + y) as follows.…”

Improved bounds of Mitrinović–Adamović-type inequalities by using two-parameter functions

“…are known as Mitrinović-Adamović inequalities (see [1][2][3]). Many references [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23] have discussed the problems related to Eq. ( 1), such as the following power exponential inequality obtained by Nishizawa in [4] sin x x…”

Optimal bounds for the sine and hyperbolic tangent means IV

“…For other bounds of Seiffert-like means by the arithmetic and centroidal means, see e.g. [7,8,17,20]. Similar subjects were considered also in [2][3][4][5][6]10,[13][14][15][16]18,21].…”