Optimal Bounds for Neuman Mean Using Arithmetic and Centroidal Means

Authors:

Open Access

We present the best possible parameters

α_{1}, α_{2}, β_{1}, β_{2} \in R

and

α_{3}, β_{3} \in (1 / 2,1)

such that the double inequalities

α_{1} A (a, b) + (1 - α_{1}) C (a, b) < N_{Q A} (a, b) < β_{1} A (a, b) + (1 - β_{1}) C (a, b),

A^{α_{2}} (a, b) C^{1 - α_{2}} (a, b) < N_{Q A} (a, b) < A^{β_{2}} (a, b) C^{1 - β_{2}} (a, b),

and

C [α_{3} a + (1 - α_{3}) b, α_{3} b + (1 - α_{3}) a] < N_{Q A} (a, b) < C [β_{3} a + (1 - β_{3}) b, β_{3} b + (1 - β_{3}) a]

hold for all

a, b > 0

with

a \neq b

and give several sharp inequalities involving the hyperbolic and inverse hyperbolic functions. Here,

N (a, b)

A (a, b)

Q (a, b)

, and

C (a, b)

are, respectively, the Neuman, arithmetic, quadratic, and centroidal means of

a

and

b

, and

N_{Q A} (a, b) = N [Q (a, b), A (a, b)]

National Natural Science Foundation of China (11371125, 61374086, LY13A010004, 12C0577, XKT-15G17)
National Natural Science Foundation of China (11371125, 61374086, LY13A010004, 12C0577, XKT-15G17)
Natural Science Foundation of Zhejiang Province (11371125, 61374086, LY13A010004, 12C0577, XKT-15G17)
Natural Science Foundation of Hunan Province (11371125, 61374086, LY13A010004, 12C0577, XKT-15G17)
Natural Science Foundation of Zhejiang Province (11371125, 61374086, LY13A010004, 12C0577, XKT-15G17)

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