Abstract
In this paper, we establish sharp Shafer-Fink type inequalities for Gauss lemniscate functions.
MSC:26D07.
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1 Introduction and definitions
In geometry, the lemniscate of Bernoulli is a plane curve defined by two given points and , known as foci, at distance 2a from each other as the locus of points P so that . This gives the equation . In polar coordinates , the equation becomes . The arc length from the origin to a point on the Bernoulli lemniscate is given by the function
where arcslx is called the arc lemniscate sine function studied by CF Gauss in 1797-1798. Another lemniscate function investigated by Gauss is the hyperbolic arc lemniscate sine function, defined as
The functions (1.1) and (1.2) can be found in [[1], p.259], [[2], (2.5)-(2.6)], [3–7] and [[8], Ch. 1].
Following Neuman [3], Gauss’ arc lemniscate tangent and the hyperbolic arc lemniscate tangent functions are defined by
and
respectively.
For , it is known in the literature that
The first and second inequalities in equation (1.5) were established by Shafer (see, e.g., [[9], p.247]), while the third inequality was proved by Fink [10]. In recent years, Shafer-Fink’s inequalities have attracted much attention of the mathematical community. By using the λ-method of Mitrinović and Vasić [9], Malešević [11] improved the upper bound for arcsinx and established the following inequality: For ,
In [12–15], other upper bounds for arcsinx were established: For ,
In [16], Pan and Zhu gave some further generalizations of these results and obtained two new Shafer-Fink type double inequalities. In [17], Zhu provided a solution to an open problem posed by Oppenheim in [18]. At the same time, some Shafer-Fink inequalities were deduced from the solution of Oppenheim’s problem. Chen and Cheung [19] provided a laconic proof to Oppenheim’s problem. Recently, Qi and Guo [20, 21] presented a sharpening and generalizations of Shafer-Fink’s inequality.
Related to the inverse sine inequality, the inverse tangent inequality is also of much interest. In the literature, we have
The first inequality in equation (1.8) was presented without proof by Shafer [22]. Three proofs of it were later given in [23]. The second inequality in equation (1.8) can be found in, e.g., [[24], p.288]. Shafer’s inequality (1.8) was recently sharpened and generalized by Qi et al. in [25]. For each , Chen et al. [26] determined the largest number and the smallest number such that the inequalities
are valid for all . Zhu [[27], Theorems 1.9 and 1.10] established Shafer-Fink type inequalities for the inverse hyperbolic sine function.
Recently, numerous inequalities have been given for the lemniscate functions. For example, Neuman [5] proved the following inequalities:
and
for .
Chen [28, 29] established Wilker and Huygens type inequalities for Gauss lemniscate functions. For example, Chen [28] proved that for ,
with the best possible constant . Chen [29] proved that for ,
In this paper, we establish sharp Shafer-Fink type inequalities for Gauss lemniscate functions.
The following lemma is required in our present investigation.
Let , and let be continuous on , differentiable on . Let on . If is increasing (decreasing) on , then so are
If is strictly monotone, then the monotonicity in the conclusion is also strict.
Remark 1.1 A generalization of the familiar trigonometric and hyperbolic functions was described by Lindqvist [33]. The generalized p-trigonometric functions occur as an eigenfunction of the Dirichlet problem for the one-dimensional p-Laplacian. Recently, the p-trigonometric functions have been studied extensively, see for example [34–37] and their references. Very recently, Takeuchi [38] (see also [39]) introduced the -trigonometric functions that coincide with the p-trigonometric functions for and are connected with the Dirichlet problem for the -Laplacian. These -trigonometric functions have been the subject of intense investigations (see, for example, [35, 39–42]). For the function is defined in [38, 39] by
Similarly, for the function is defined by [40]
Clearly,
2 Main results
Theorem 2.1 For ,
with the best possible constants
Here
is the beta function.
Proof For , let
where
Then,
Differentiation yields
where
Motivated by the investigations in [12], we are in a position to prove for . Let
where μ is constant determined with limit:
Using Maple we determine Taylor approximation for the function by the polynomial of the fourth order:
which has a bound of absolute error
for values . It is true that
and
Hence, for it is true that and therefore and for . Therefore, the function is strictly increasing on . By Lemma 1.1, the function
is strictly increasing on . And hence,
for . By rearranging terms in the last expression, Theorem 2.1 follows. □
Theorem 2.2 For ,
with the best possible constants
Here denotes the beta function.
Proof For , let
where
Then,
Differentiation yields
where
Motivated by the investigations in [12], we are in a position to prove for . Let
where λ is a constant determined by the limit
Using Maple we determine a Taylor approximation for the function by the polynomial of fourth order:
which has a bound of the absolute error of
for values . It is true that
and
Hence, for it is true that and therefore and for . Therefore, the function is strictly increasing on . By Lemma 1.1, the function
is strictly increasing on . And hence,
for . By rearranging terms in the last expression, Theorem 2.2 follows. □
Theorem 2.3 For ,
and
with the best possible constants
and
Here denotes the beta function.
Proof For , let
Differentiation yields
where
Elementary calculation shows that
Hence, and for . Therefore, the function is strictly increasing on . And hence,
for . Hence, inequality (2.5) holds with the best possible constants given in equation (2.7).
For , let
Differentiation yields
where
Elementary calculation shows that
where
We claim that for . By an elementary change of variable
we find that
where
Obviously, for . This proves the claim.
Hence, for . This implies that and for . Therefore, the function is strictly decreasing on . And hence,
for . Hence, inequality (2.6) holds with the best possible constants given in equation (2.8). □
Remark 2.1 (i) There is no strict comparison between the two lower bounds in equations (2.5) and (2.6). Also, there is no strict comparison between the two upper bounds in equations (2.5) and (2.6).
There is no strict comparison between the two upper bounds in equations (1.10) and (2.5).
-
(iii)
By two elementary changes of variable,
we find that
Hence, the upper bound in equation (2.6) is sharper than the one in equation (1.10). There is no strict comparison between the two lower bounds in equations (1.10) and (2.6).
Theorem 2.4 For ,
with the best possible constants
Here denotes the beta function.
Proof The inequality (2.10) is obtained by considering the function defined by
Differentiation yields
where
Elementary calculation shows that
where
We claim that for . By an elementary change of variable
we find that
where
Obviously, for . This proves the claim.
Hence, and for . Therefore, for , and we have
Hence, the inequality (2.10) holds with the best possible constants given in equation (2.11). □
Remark 2.2 Inequality (1.11) is sharper than inequality (2.10).
Theorem 2.5 For ,
with the best possible constants
Here denotes the beta function.
Proof The inequality (2.13) is obtained by considering the function defined by
Differentiation yields
where
Elementary calculation shows that
Hence, for . Therefore, for , and we have
Hence, the inequality (2.13) holds with the best possible constants given in equation (2.14). □
Theorem 2.6 For ,
and
with the best possible constants
and
Proof For , let
Differentiation yields
where
Elementary calculation shows that
where
We claim that for . By an elementary change of variable
we find that
where
Obviously, for . This proves the claim.
Hence, for . This implies that and for . Therefore, the function is strictly increasing on . And hence,
for . Hence, inequality (2.15) holds with the best possible constants given in equation (2.17).
For , let
Differentiation yields
where
Elementary calculation shows that
which implies that and for . Therefore, the function is strictly decreasing on . And hence,
for . Hence, inequality (2.16) holds with the best possible constants given in equation (2.18). □
Remark 2.3 There is no strict comparison between the two lower bounds in equations (2.15) and (2.16). Also, there is no strict comparison between the two upper bounds in equations (2.15) and (2.16).
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Deng, JE., Chen, CP. Sharp Shafer-Fink type inequalities for Gauss lemniscate functions. J Inequal Appl 2014, 35 (2014). https://doi.org/10.1186/1029-242X-2014-35
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DOI: https://doi.org/10.1186/1029-242X-2014-35