Abstract
In the article, we provide several sharp upper and lower bounds for two Sándor–Yang means in terms of combinations of arithmetic and contra-harmonic means.
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1 Preliminaries
Let \(a, b>0\) with \(a\neq b\). Then the arithmetic mean \(A(a,b)\) [1–4], the quadratic mean \(Q(a,b)\) [5], the contra-harmonic mean \(C(a, b)\) [6–9], the Neuman–Sándor mean \(NS(a, b)\) [10–12], the second Seiffert mean \(T(a, b)\) [13, 14], and the Schwab–Borchardt mean \(SB(a, b)\) [15, 16] of a and b are defined by
respectively, where \(\sinh^{-1}(x)=\log(x+\sqrt{x^{2}+1})\) and \(\cosh^{-1}(x)=\log(x+\sqrt{x^{2}-1})\) are respectively the inverse hyperbolic sine and cosine functions. The Schwab–Borchardt mean \(SB(a,b)\) is strictly increasing, non-symmetric and homogeneous of degree one with respect to its variables. It can be expressed by the degenerated completely symmetric elliptic integral of the first kind [17]. Recently, the Schwab–Borchardt mean has attracted the attention of many researchers. In particular, many remarkable inequalities for the Schwab–Borchardt mean and its generated means can be found in the literature [18–38].
Let \(X(a,b)\) and \(Y(a,b)\) denote symmetric bivariate means of a and b. Then Yang [39] introduced the Sándor–Yang mean
and presented the explicit formulas for \(R_{QA}(a,b)\) and \(R_{AQ}(a,b)\) as follows:
Very recently, the bounds involving the Sándor–Yang means have been the subject of intensive research. Numerous interesting results and inequalities for \(R_{QA}(a,b)\) and \(R_{AQ}(a,b)\) can be found in the literature [40–42].
Neuman [43] established the inequality
for \(a,b > 0\) with \(a\neq b\).
In [44], Xu proved that the double inequalities
hold for all \(a,b > 0\) with \(a\neq b\) if and only if \(\alpha_{1}\leq (1+\sqrt{2})^{\sqrt{2}}/e-1=0.2794\ldots\) , \(\beta_{1}\geq 1/3\), \(\alpha_{2}\leq \sqrt{2}e^{\pi/4-1}-1=0.1410\ldots\) and \(\beta_{2}\geq 1/6\).
From (1.6) and (1.7), together the well-known inequalities
we clearly see that
for all \(a, b >0\) with \(a\neq b\).
The main purpose of this paper is to find the best possible parameters \(\alpha_{i}, \beta_{i}\in (0, 1)\) \((i=1, 2, 3, 4)\) such that the double inequalities
hold for all \(a, b>0\) with \(a\neq b\).
2 Lemmas
In order to prove our main results, we need several lemmas, which we present in this section.
Lemma 2.1
(see [45])
Let \(a, b\in \mathbb{R}\) with \(a< b\), \(f, g: [a, b]\mapsto \mathbb{R}\) be continuous on \([a, b]\) and differentiable on \((a, b)\), and \(g^{\prime}(x)\neq 0\) on \((a, b)\). If \(f^{\prime}(x)/g^{\prime}(x)\) is increasing (decreasing) on \((a, b)\), then so are the functions
If \(f^{\prime}(x)/g^{\prime}(x)\) is strictly monotone, then the monotonicity in the conclusion is also strict.
Lemma 2.2
(see [46])
Let \(A(t)=\sum_{k=0}^{\infty}a_{k}t^{k}\) and \(B(t)=\sum_{k=0}^{\infty }b_{k}t^{k}\) be two real power series converging on \((-r,r)\) (\(r>0\)) with \(b_{k}>0\) for all k. If the non-constant sequence \(\{a_{k}/b_{k}\}_{k=0}^{\infty}\) is increasing (decreasing) for all k, then the function \(t\mapsto A(t)/B(t)\) is strictly increasing (decreasing) on \((0,r)\).
Lemma 2.3
The function
is strictly increasing from \((0, \log(1+\sqrt{2})\) onto \((1/3, [\sqrt{2}\log(1+\sqrt{2})-1]/\log 2)\).
Proof
Let \({\phi _{1}} ( x ) = x\coth ( x ) - 1\), \({\phi _{2}} ( x ) = 2\log [ {\cosh ( x )} ]\). Then elaborate computations lead to
Let
Then
and
for all \(n \ge 0\).
It follows from Lemma 2.2 and (2.2)–(2.5) that \({\phi '_{1}} ( x )/{\phi '_{2}} ( x )\) is strictly increasing on \(( {0,\log ( {1 + \sqrt {2} } )} )\).
Note that
Therefore, Lemma 2.3 follows from Lemma 2.1, (2.1), and (2.6) together with the monotonicity of \({\phi '_{1}} ( x )/{\phi '_{2}} ( x )\). □
Lemma 2.4
The function
is strictly increasing from \((0, \pi/4)\) onto \((1/6, 1/2-(4-\pi)(4\log 2))\).
Proof
Let \({\varphi _{1}} ( x ) = \log \sec ( x ) + x\cot ( x ) - 1\), \({\varphi _{2}} ( x ) = 2\log [ {\sec ( x )} ]\), \({\varphi _{3}} ( x ) = \sin ( x ) - x\cos ( x )\), and \({\varphi _{4}} ( x ) = 2{\sin ^{3}} ( x )\). Then elaborate computations lead to
and
It is well known that the function \(x \to \sin ( x )/x\) is strictly decreasing on \(( {0,\pi /2} )\), hence equation (2.9) leads to the conclusion that the function \({\varphi '_{3}} ( x )/{\varphi '_{4}} ( x )\) is strictly increasing on \(( {0,\pi /4} )\).
Note that
Therefore, Lemma 2.4 follows from Lemma 2.1 and (2.7)–(2.9) together with the monotonicity of \({\varphi '_{3}} ( x )/{\varphi '_{4}} ( x )\). □
Lemma 2.5
Let \(p \in ( {0,1} )\) and
Then the following statements are true:
-
(1)
If \(p = 3/10\), then \(f ( x ) > 0\) for all \(x \in ( {1,\sqrt[6]{2}} )\);
-
(2)
If \(p = 3 [ {{{ ( {1 + \sqrt {2} } )}^{\sqrt {2} }}/e - \sqrt[3]{2}} ]/ ( {4 - 3\sqrt[3]{2}} ) = 0.2663 \dots\) , then there exists \({\lambda _{0}}( = 1.0808 \dots ) \in ( {1,\sqrt[6]{2}} )\) such that \(f ( x ) < 0\) for \(x \in ( {1,{\lambda _{0}}} )\) and \(f ( x ) > 0\) for \(x \in ( {{\lambda _{0}},\sqrt[6]{2}} )\).
Proof
Part \((1)\) follows easily from
for all \(x \in ( {1,\sqrt[6]{2}} )\) if \(p = 3/10\).
For part \((2)\), if \(p = 3 [ {{{ ( {1 + \sqrt {2} } )}^{\sqrt {2} }}/e - \sqrt[3]{2}} ]/ ( {4 - 3\sqrt[3]{2}} )\), then numerical computations lead to
It follows from (2.11) and (2.14) that
for all \(x \in ( {1,\sqrt[6]{2}} )\).
Therefore, part \((2)\) follows easily from (2.12), (2.13), (2.15), and the numerical results \(f ( {1.0808} ) < 0\) and \(f ( {1.0809} ) > 0\). □
Lemma 2.6
Let \(p \in ( {0,1} )\) and
Then the following statements are true:
-
(1)
If \(p = 12/25\), then \(g ( x ) > 0\) for all \(x \in ( {1,\sqrt[6]{2}} )\);
-
(2)
If \(p = 6 [ {\sqrt {2} {e^{\pi/4- 1}} - \sqrt[6]{2}} ]/ ( {7 - 6\sqrt[6]{2}} ) = 0.4210 \dots\) , then there exists \({\mu _{0}}( = 1.0577 \dots ) \in ( {1,\sqrt[6]{2}} )\) such that \(g ( x ) < 0\) for \(x \in ( {1,{\mu _{0}}} )\) and \(g ( x ) > 0\) for \(x \in ( {{\mu _{0}},\sqrt[6]{2}} )\).
Proof
Part \((1)\) follows easily from
for all \(x \in ( {1,\sqrt[6]{2}} )\) if \(p = 12/25\).
For part \((2)\), if \(p =6[\sqrt{2}e^{\pi/4-1}-\sqrt[6]{2}]/(7-6\sqrt[6]{2}) = 0.4210 \ldots\) , then numerical computations lead to
It follows from (2.16) and (2.19) that
for \(x \in ( {1,\sqrt[6]{2}} )\).
Therefore, part \((2)\) follows easily from (2.17), (2.18), and (2.20) together with the numerical results \(g ( {1.0577} ) < 0\) and \(g ( {1.0578} ) > 0\). □
3 Main results
We are now in a position to state and prove our main results.
Theorem 3.1
The double inequality
holds for all \(a,b>0 \) with \(a\ne b \) if and only if \({\alpha _{1}} \le 1/3 \) and \({\beta _{1}} \ge [ {\sqrt {2} \log ( {1 + \sqrt {2} } ) - 1} ]/\log 2 \).
Proof
Clearly, inequality (3.1) can be rewritten as
Since \(A(a,b)\), \({R_{QA}} ( {a,b} )\), and \(C(a,b)\) are symmetric and homogenous of degree one, we assume that \(a > b > 0\). Let \(v = ( {a - b} )/ ( {a + b} ) \in ( {0,1} )\). Then from (1.1), (1.2), and (1.4) we know that inequality (3.2) is equivalent to
Let \(x =\sinh^{-1} ( v )\). Then \(x \in ( {0,\log ( {1 + \sqrt {2} } )} )\) and
Therefore, inequality (3.1) holds for all \(a,b > 0\) with \(a \ne b\) if and only if \({\alpha _{1}} \le 1/3\) and \({\beta _{1}} \ge [ {\sqrt {2} \log ( {1 + \sqrt {2} } ) - 1} ]/\log 2\) follows from (3.2)–(3.4) and Lemma 2.3. □
Theorem 3.2
The double inequality
holds for all \(a,b>0 \) with \(a\ne b \) if and only if \({\alpha _{2}} \le 1/6 \) and \({\beta _{2}} \ge 1/2 - ( {4 - \pi } )/ ( {4\log 2} ) = 0.1903 \dots \) .
Proof
Clearly, inequality (3.5) can be rewritten as
Since \(A(a,b)\), \({R_{AQ}} ( {a,b} )\), and \(C(a,b)\) are symmetric and homogenous of degree one, we assume that \(a > b > 0\). Let \(v = ( {a - b} )/ ( {a + b} ) \in ( {0,1} )\). Then from (1.1), (1.3), and (1.5) we see that inequality (3.6) is equivalent to
Let \(x = \arctan ( v )\). Then \(x \in ( {0,\pi /4} ) \) and
Therefore, inequality (3.5) holds for all \(a,b > 0\) with \(a \ne b\) if and only if \({\alpha _{2}} \le 1/6 \) and \({\beta _{2}} \ge 1/2 - ( {4 - \pi } )/ ( {4\log 2} ) = 0.1903 \dots \) follows from (3.6)–(3.8) and Lemma 2.4. □
Theorem 3.3
The double inequality
holds for all \(a,b>0 \) with \(a\ne b \) if and only if \({\alpha _{3}} \le 3 [ {{{ ( {1 + \sqrt {2} } )}^{\sqrt {2} }}/e - \sqrt[3]{2}} ]/ ( {4 - 3\sqrt[3]{2}} ) = 0.2663 \dots \) and \({\beta _{3}} \ge 3/10 \).
Proof
Since \({R_{QA}} ( {a,b} )\), \(A(a,b)\), and \(C(a,b)\) are symmetric and homogenous of degree one, without loss generality, we assume that \(a > b > 0\). Let \(v = ( {a - b} )/ ( {a + b} ) \), \(x = \sqrt[6]{{1 + {v^{2}}}}\), and \(p \in ( {0,1} )\). Then \(v \in ( {0,1} )\), \(x \in ( {1,\sqrt[6]{2}} ) \), and (1.1), (1.2), and (1.4) lead to
Let
Then simple computations lead to
where
where \(f ( x ) \) is defined as in Lemma 2.5.
We divide the proof into four cases.
Case 1 \(p = 3/10 \). Then it follows from (3.9)–(3.14) and Lemma 2.5(1) that
Case 2 \(0 < p < 3/10\). Let \(v > 0 \) and \(v \to {0^{+} } \). Then power series expansion leads to
Equations (3.9), (3.10), and (3.15) lead to the conclusion that there exists \(0 < {\delta _{1}} < 1\) such that
for all \(a > b > 0\) with \(( {a - b} )/ ( {a + b} ) \in ( {0,{\delta _{1}}} )\).
Case 3 \(p = 3 [ {{{ ( {1 + \sqrt {2} } )}^{\sqrt {2} }}/e - \sqrt[3]{2}} ]/ ( {4 - 3\sqrt[3]{2}} )\). Then (3.13) leads to
Let \({\lambda _{0}} = 1.0808 \dots \) be the number given in Lemma 2.5(2). Then we divide the discussion into two subcases.
Subcase 1 \(x \in ( {1,{\lambda _{0}}} ]\). Then \({F_{1}} ( x ) > 0\) for \(x \in ( {1,{\lambda _{0}}} ]\) follows easily from (3.13) and (3.14) together with Lemma 2.5(2).
Subcase 2 \(x \in ( {{\lambda _{0}},\sqrt[6]{2}} )\). Then Lemma 2.5(2) and (3.14) lead to the conclusion that \({F_{1}} ( x )\) is strictly decreasing on the interval \([ {{\lambda _{0}},\sqrt[6]{2}} )\). Then, from (3.16) and Subcase 1, we know that there exists \({\lambda _{1}} \in ( {{\lambda _{0}},\sqrt[6]{2}} )\) such that \({F_{1}} ( x ) > 0\) for \(x \in [ {{\lambda _{0}},{\lambda _{1}}} )\) and \({F_{1}} ( x ) < 0\) for \(x \in ( {{\lambda _{1}},\sqrt[6]{2}} )\).
It follows from Subcases 1 and 2 together with (3.12) that \(F ( x )\) is strictly increasing on \(( {1,{\lambda _{1}}} ]\) and strictly decreasing on \([ {{\lambda _{1}},\sqrt[6]{2}} )\). Therefore,
follows from (3.9)–(3.11) and (3.16) together with the piecewise monotonicity of \(F ( x )\).
Case 4 \(3 [ {{{ ( {1 + \sqrt {2} } )}^{\sqrt {2} }}/e - \sqrt[3]{2}} ]/ ( {4 - 3\sqrt[3]{2}} ) < p < 1\). Then (3.11) leads to
Equations (3.9) and (3.10) together with inequality (3.17) imply that there exists \(0 <\delta_{1}^{\ast} < 1\) such that
for all \(a > b > 0\) with \(( {a - b} )/ ( {a + b} ) \in ( {1 -\delta_{1}^{\ast} ,1} )\). □
Theorem 3.4
The double inequality
holds for all \(a,b>0 \) with \(a\ne b \) if and only if \({\alpha _{4}} \le 6 [ {\sqrt {2} {{\mathrm{e}} ^{ ( {\pi /4 - 1} )}} - \sqrt[6]{2}} ]/ ( {7 - 6\sqrt[6]{2}} ) = 0.4210 \dots \) and \({\beta _{4}} \ge 12/25 \).
Proof
Since \({R_{AQ}} ( {a,b} )\), \(A(a,b)\), and \(C(a,b)\) are symmetric and homogenous of degree one, without loss generality, we assume that \(a > b > 0\). Let \(v = ( {a - b} )/ ( {a + b} ) \), \(x = \sqrt[6]{{1 + {v^{2}}}}\), and \(p \in ( {0,1} )\). Then \(v \in ( {0,1} )\), \(x \in ( {1,\sqrt[6]{2}} ) \) and (1.1), (1.3), and (1.5) lead to
Let
Then simple computations lead to
where
where \(g ( x ) \) is defined as in Lemma 2.6.
We divide the proof into four cases.
Case 1 \(p = 12/25 \). Then it follows from (3.18)–(3.23) and Lemma 2.6(1) that
Case 2 \(0 < p < 12/25\). Let \(v > 0 \) and \(v \to {0^{+} } \), then power series expansion leads to
Equations (3.18), (3.19), and (3.24) lead to the conclusion that there exists \(0 < {\delta _{2}} < 1\) such that
for all \(a > b > 0\) with \(( {a - b} )/ ( {a + b} ) \in ( {0,{\delta _{2}}} )\).
Case 3 \(p = 6 [ {\sqrt {2} {{\mathrm{e}} ^{ ( {\pi /4 - 1} )}} - \sqrt[6]{2}} ]/ ( {7 - 6\sqrt[6]{2}} )\). Then, from (3.20) and (3.22) together with numerical computations, we get
Let \({\mu _{0}} = 1.0577 \dots \) be the number given in Lemma 2.6(2). Then we divide the discussion into two subcases.
Subcase 1 \(x \in ( {1,{\mu _{0}}} ] \). Then \({G_{1}} ( x ) > 0\) for \(x \in ( {1,{\mu _{0}}} ]\) follows easily from (3.22) and (3.23) together with Lemma 2.6(2).
Subcase 2 \(x \in ( {{\mu _{0}},\sqrt[6]{2}} )\). Then Lemma 2.6(2) and (3.23) lead to the conclusion that \({G_{1}} ( x )\) is strictly decreasing on the interval \([ {{\mu _{0}},\sqrt[6]{2}} )\). Then, from (3.25) and Subcase 1, we know that there exists \({\mu _{1}} \in ( {{\mu _{0}},\sqrt[6]{2}} )\) such that \({G_{1}} ( x ) > 0\) for \(x \in [ {{\mu _{0}},{\mu _{1}}} )\) and \({G_{1}} ( x ) < 0\) for \(x \in ( {{\mu _{1}},\sqrt[6]{2}} )\).
It follows from Subcases 1 and 2 together with (3.21) that \(G ( x )\) is strictly increasing on \(( {1,{\mu _{1}}} ]\) and strictly decreasing on \([ {{\mu _{1}},\sqrt[6]{2}} )\). Therefore,
follows from (3.18)–(3.20) and (3.25) together with the piecewise monotonicity of \(G ( x )\).
Case 4 \(6 [ {\sqrt {2} {{\mathrm{e}} ^{ ( {\pi /4 - 1} )}} - \sqrt[6]{2}} ]/ ( {7 - 6\sqrt[6]{2}} ) < p < 1 \). Then (3.21) leads to
Equations (3.18) and (3.19) together with inequality (3.26) imply that there exists \(0 < \delta_{2}^{\ast}< 1\) such that
for all \(a > b > 0\) with \(( {a - b} )/ ( {a + b} ) \in ( {1 - \delta_{2}^{\ast}, 1} )\). □
4 Results and discussion
In this paper, we provide the optimal upper and lower bounds for the Sándor–Yang means \(R_{QA}(a,b)\) and \(R_{AQ}(a,b)\) in terms of combinations of the arithmetic mean \(A(a,b)\) and the contra-harmonic mean \(C(a,b)\). Our approach may have further applications in the theory of bivariate means.
5 Conclusion
In the article, we find several best possible bounds for the Sándor–Yang means \(R_{QA}(a,b)\) and \(R_{AQ}(a,b)\). These results are improvements and refinements of the previous results.
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The research was supported by the Natural Science Foundation of China (Grants Nos. 61673169, 61374086, 11371125, 11401191), the Tianyuan Special Funds of the National Natural Science Foundation of China (Grant No. 11626101) and the Natural Science Foundation of the Department of Education of Zhejiang Province (Grant No. Y201635325).
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Xu, HZ., Chu, YM. & Qian, WM. Sharp bounds for the Sándor–Yang means in terms of arithmetic and contra-harmonic means. J Inequal Appl 2018, 127 (2018). https://doi.org/10.1186/s13660-018-1719-6
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DOI: https://doi.org/10.1186/s13660-018-1719-6