Inequalities for the generalized trigonometric and hyperbolic functions with two parameters

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1 Available online at J. Nonlinear Sci. Al. 8 5, Research Article Inequalities for the generalized trigonometric and hyerbolic functions with two arameters Li Yin a,, Li-Guo Huang a a Deartment of Mathematics, Binzhou University, Binzhou City, 5663 Shandong Province, China. Abstract In this aer, we resent some integral identities and inequalities of, q comlete ellitic integrals, and rove some inequalities for the generalized trigonometric and hyerbolic functions with two arameters. c 5 All rights reserved. Keywords: comlete ellitic integrals, inequality, generalized trigonometric function, generalized hyerbolic function, Fubini theorem. MSC: 33B, 33E5.. Introduction The generalized trigonometric and hyerbolic functions deending on a arameter > were studied by P. Lindqvist in a highly cited aer see [3]. Motivated by this work, many authors have studied the equalities and inequalities related to generalized trigonometric and hyerbolic functions in [5, 7, ]. Recently, in [7], S. Takeuchi has investigated the, q trigonometric functions deending on two arameters and in which the case of q coincides with the function of Lindqvist, and for q they coincide with familiar elementary functions. For <, q < and, the arc sine may be generalized as arcsin,q t q dt. / and π,q arcsin,q t q dt.. / Corresonding author addresses: yinli_79@63.com Li Yin, liguoh3@sina.com Li-Guo Huang Received 4--5

2 L. Yin, L. G. Huang, J. Nonlinear Sci. Al. 8 5, The inverse of arcsin,q on [ ], π,q is called the generalized, q sine function, denoted by sin,q, and may be etended to,. In the same way, we can define the generalized, q cosine function, the generalized, q tangent function and their inverses. Their definitions and formulas can be found in [9, ]. Similarly, we can define the inverse of the generalized, q hyerbolic sine function as follows. arcsinh,q + t q dt.3 / and also other corresonding, q hyerbolic functions. In [6], B. A. Bhayo and M. Vuorinen establish some inequalities and resent a few conjectures for the, q functions. Very recently, a conjecture osed in [6] was verified in []. Legendre s comlete ellitic integrals of the first and second kind are defined for real numbers < r < by and κr εr π/ π/ r sin t dt r sin tdt dt.4 t r t r t dt.5 t resectively. The comlete ellitic integrals have many alications in several mathematical branches as well as in engineering and hysics. Motivated by roblems in otential theory and in the theory of quasi-conformal maings, many mathematicians obtain monotonicity and conveity theorems of certain combinations of κr and εr. See [,, 3, 4, 8,, 5, 8]. In the second section of the aer, we define, q comlete ellitic integrals, and rove some integral identities and inequalities. In the final section, we obtain some inequalities related to generalized trigonometric and hyerbolic functions with two arameters.. Some roerties related to, q-comlete ellitic integrals Definition.. For all, q, and r,, the following the first and second kind of, q-comlete ellitic integrals are defined by { κ,q r π,q/ r q sin q,q θ / dθ κ,q π,q, κ,q,. and { resectively. ε,q r π,q/ r q sin q,q θ / dθ ε,q π,q, ε,q.. Remark.. For q, they coincide with the first and second kind of comlete ellitic integrals. Lemma.3 [9]. For all, q, + and all θ, π,q ], then sin,q θ π,q θ Theorem.4. For all, q,, r, and θ, π,q κ,q rdr π,q/..3 θ dθ..4 sin,q θ

3 L. Yin, L. G. Huang, J. Nonlinear Sci. Al. 8 5, Proof. The substitution t r turns the identity into Setting θ arcsin,q, we have From.7, it follows that by using Fubini theorem. arcsin,q t q dt.5 / arcsin,q r q q dr.6 / θ sin,q θ r q sin q dr..7,q θ / π,q/ θ sin,q θ dθ π,q/ π,q/ Corollary.5. For all, q,, r, r q sin q,q θ / dr r q sin q dθ,q θ / dθ dr κ,q rdr.8 π,q κ,q rdr π,q 4..9 Proof. Using Lemma.3 and Theorem.4, we easily obtain the inequality.9. Theorem.6. For all, q,, r, and θ, π,q where +. ε,q rdr Proof. By definite integration by art, we have So, we have t q / dt q / + q + q + q + q t q / dt + q q / q + + q The substitution t r turns. into Setting θ arcsin,q, we have t q / dt q r q q / dr + q q / q + + q r q sin q,q θ / dr + q cos,q θ + κ,qrdr,. t q / dt.. t q / dt.. q + q Similar to the roof of Theorem.4, we easily obtain. by using Fubini theorem. r q q / dr..3 r q sin q,q θ / dr..4

4 L. Yin, L. G. Huang, J. Nonlinear Sci. Al. 8 5, Theorem.7. For all, q,, r, ε,qr q ε,q r κ r,qr.5 where +. Proof. For all, q,, r, ε,qr q π,q/ π,q/ r q sin q,q θ / r q sin q,q θdθ q r q sin q,q θ / r q sin q,q θ dθ r q ε,q r κ r,qr. Lemma.8 [4]. Let f, g be integrable functions in [a, b], both increasing or both decreasing. Then b a b a fgd b a b a fd b a b a gd..6 If one of the functions f or g is nonincreasing and the other nondecreasing, then the inequality in.6 is reversed. Lemma.9. For all, q, and θ, π,q π,q/ Proof. Putting t sin,q θ and t q u, we have sin q,q θdθ q..7 π,q/ sin q,q θdθ t q t q / dt q B, Γ / q Γ / q. Lemma.. For all, q, and θ, π,q where A, q, r rq / r q / r π,q/ u q q/ u q / du. sin q,q θ r q sin q dθ A, q, r,.8,q θ /

5 L. Yin, L. G. Huang, J. Nonlinear Sci. Al. 8 5, Proof. Putting t cos,q θ and t rq r q q u q u q, we have π,q/ rq / r q / sin q,q θ r q sin q dθ,q θ / t r q + r q t r / dt u q q/ u q / du. Theorem.. For all, q,, r, and θ, π,q π,q arcsin,q r r κ,q r qπ,qa, q, r..9 Proof. It is easily known that the functions fθ r q sin q,q θ / and gθ cos,q θ are increasing and decreasing in, π,q. Using Tchebychef s inequality.6 in Lemma.8 and substitution of variable t sin,q θ, rt u, then κ,q r π,q π,q π,q π,q π,q/ r cos,q θ r q sin q dθ,q θ / dt r q t q / u q / r du arcsin,q r. r So, the roof of the first inequality is comleted. Similarly, Putting fθ r q sin q,q θ / and gθ sin,q q θ in Lemma.8 and alying Lemma.9 and., we easily obtain the second inequality. Thus, we accomlished the inequalities.9. Putting fθ r q sin q,q θ / and gθ cos,q θ or gθ sin q,q θ in Lemma.8, we easily obtain the following theorem. Theorem.. For all, q,, r, and θ, π,q where λ, q, r rq r r q / π,q λ, q, r r ε,q r π,q u q q / du and µ, q, r r u q uq / du. 3. Some Inequalities, q trigonometric and hyerbolic functions µ, q, r,. r Lemma 3.. Let the nonemty number set D,, the maing f : D J, is a bijective function. Assume that function g is ositive increasing and f g D, k > is strictly increasing. If f y for all D, then gy fgf y, where f : J D denotes the inverse function of f;

6 L. Yin, L. G. Huang, J. Nonlinear Sci. Al. 8 5, If f y for all D, then gy fgf y. Proof. The roof of Lemma is similar to Theorem. of [6]. Here we omit the detail. Lemma 3. [6]. For all, q,,, { + q π,q +q < arcsin,q < min, q /+q}, / + L, q, < q arcsinh,q < / + U, q,, q where L, q, ma { U, q, q q +q+ q q + q q/q+., + q / q++q q q+ /q }, Theorem 3.3. For all, q,, and, e arcsin,q esin,q + q +q + q +q Proof. Setting g e and f arcsin,q,, in Lemma 3., we have f g e q / arcsin,q. In fact, since the function q / is strictly increasing, we easily obtain arcsin,q t q / dt q / < q /.. 3. So, f g imlies that the function f g is increasing for,. Taking y + q alying Lemma 3., we have y f. By using Lemma 3., we easily obtain inequality 3.. Theorem 3.4. For all, q,, and, ξ e arcsinh,q e sinh,q + q /U,q, where ξ is an unique ositive root of equation + q /. Proof. Define h + q /. A direct comutation yields h + q / + q q + q / <. Thus, the function h is decreasing on,. Setting g e and +q and + q / U, q,, 3. f arcsinh,q,, ξ

7 L. Yin, L. G. Huang, J. Nonlinear Sci. Al. 8 5, in Lemma 3., we have f g e + q / arcsinh,q > e + q / + q / e + q /. Using Lemma 3. and Lemma 3., we easily obtain the inequality 3.. Theorem 3.5. For >, q > and, q cos,q d > q Proof. Putting t arcsin,q, the left integral of 3.3 becomes sin,q q d. 3.3 q cos,q π,q/ d q cos q,q sin,q tdt. 3.4 Similarly, taking t arccos,q, the right hand side of 3.3 is reduced into sin,q q d q π,q/ Making use of the monotonicity of sin,q and cos,q, we have Thus, the inequality 3.3 is roved. sin q,q t sin,q cos,q tdt. 3.5 sin q,q t sin,q cos,q t < sin,q cos,q t < cos,q t < cos,q sin,q t. Theorem 3.6. Let >, q > satisfy / + /. For any, where B q q B q, arcsin,q arcsinh,q < q, q is incomlete beta function. t q, 3.6 / Proof. For the first inequality, it is easy to see that the function is strictly increasing and t q / +t q / is strictly decreasing for t,. Using integral eression of arcsin,q, arcsinh,q and Tchebychef s inequality, we have arcsin,q arcsinh,q t q t q q dt / / dt + t q dt / u / u / du q q B q,. q

8 L. Yin, L. G. Huang, J. Nonlinear Sci. Al. 8 5, For the second inequality, we have arcsin,q arcsinh,q t q dt / + t q dt / / / t q dt dt by using Hölder s inequality. / t q dt < / q t q / / + t q dt Remark 3.7. This aer is a revised version of reference [9]. / / / + t q dt dt Acknowledgements The first author was suorted by NSFC 44, PhD research caital of Binzhou University under grant number 3Y, a roject of Shandong Province Higher Educational Science and Technology Program J4li54, and by the Science Foundation of Binzhou University under grant BZXYL33, BZXYL4. The authors are thankful to the referee for giving valuable comments and suggestions which heled to imrove the final version of this aer. References [] H. Alzer, Shar inequalities for the comlete ellitic integrals of the first kinds, Math. Proc. Camb. Phil. Soc., 4 988, [] G. D. Anderson, P. Duren, M. K. Vamanamurthy, An inequality for comlete ellitic integrals, J. Math. Anal. Al., 8 994, [3] G. D. Anderson, S. L. Qiu, M. K. Vamanamurthy, Ellitic integrals inequalities, with alications, Constr. Aro., 4 998, [4] G. D. Anderson, M. K. Vamanamurthy, M. Vuorinen, Toics in secial functions, arxiv:7.3856v[math.ca]. [5] B. A. Bhayo, M. Vuorinen, Inequalities for eigenfunctions of the -Lalacian, Issues of Analysis : 3, 9 ages. htt://ariv.org/abs/.39 [6] B. A. Bhayo, M. Vuorinen, On generalized trigonometric functions with two arameters, J. Aro. Theory, 64, , 3. [7] B. A. Bhayo, M. Vuorinen, Power mean inequality of generalized trigonometric functions, Mat. Vesnik, to aear htt://mv.mi.sanu.ac.rs/paers/mv3_33.df. [8] Y. M. Chu, M. K. Wang, S. L. Qiu, Y. P. Jiang, Bounds for comlete ellitic integrals of second kind with alications, Comut. Math. Al., 63, [9] D. E. Edmunds, P. Gurka, J. Lang, Proerties of generalized trigonometric functions, J. Aro. Theory, 64, ,.3 [] B. -N. Guo, F. Qi, Some bounds for the comlete ellitic integrals of the first and second kind, Math. Inequal. Al., 4, [] W. D. Jiang, M. K. Wang, Y. M. Chu, Y. P. Jiang, F. Qi, Conveity of the generalized sine function and the generalized hyerbolic sine function, J. Aro. Theory, 74 3, 9., [] R. Klén, M. Vuorinen, X. H. Zhang, Inequalities for the generalized trigonometric and hyerbolic functions, J. Math. Anal. Al., 49 4, [3] P. Lindqvist, Some remarkable sine and cosine functions, Ricerche Mat., , [4] D. S. Mitrinovic, Analytic inequalities, Sringer-verlag, New York, [5] F. Qi, Z. Huang, Inequalities of the comlete ellitic integrals, Tamkang J. Math., 9 998,

9 L. Yin, L. G. Huang, J. Nonlinear Sci. Al. 8 5, [6] J. Sándor, On certain inequalities for hyerbolic and trigonometric functions, J. Math. Inequal., 7 3, [7] S. Takeuchi, Generalized Jacobian ellitic functions and their alication to bifurcation roblems associated with -Lalacian, J. Math. Anal. Al., 385, [8] M. K. Wang, Y. M. Chu, Asymtotical bounds for comlete ellitic integrals of second kind, arxiv:9. 66v[math.CA]. [9] L. Yin, L. G. Huang, Some roerties related to generalized trigonometric and hyerbolic functions with two arameters, RGMIA Res. Re. Coll. 7 4, Art.43. htt://rgmia.org/v7.h 3.7