DOI: /j3.art Issues of Analysis. Vol. 2(20), No. 2, 2013

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1 DOI: /j3.art Issues of Analysis. Vol. 2(20) No B. A. Bhayo J. Sánor INEQUALITIES CONNECTING GENERALIZED TRIGONOMETRIC FUNCTIONS WITH THEIR INVERSES Abstract. Motivate by the recent work [1] in this paper we stuy the relations of generalize trigonometric an hyperbolic functions of two parameters with their inverse functions. Key wors: Inequalities generalize trigonometric functions Eigenfunctions an Incomplete beta function Mathematical Subject Classification: 33C99 33B Introuction In [2] P. Linqvist stuie generalize trigonometric an hyperbolic functions (p-functions) for a parameter p > 1 an for p = 2 they coincie with elementary functions. These p-functions were stuie etensively see for eample [2 9] an their references. Recently these functions have been etene to (p q)-functions with two parameters p q > 1 in [10 13]. These functions coincie with the p-functions for p = q. For the historical backgroun see the bibliography of these papers. In [14] an [1] authors have stuie the inequalities involving elementary functions an their inverses. Thereafter in [14] Klén et al. stuie those results in terms of p-functions. Here we generalize those inequalities for (p q)- functions an establish ouble inequality for sin p in terms of elementary functions sin p occurs as an eigenfunction of the Dirichlet problem for the one-imensional p-laplacian see [6]. Before we formulate our main results we efine the (p q)-functions an some other notation. The increasing homeomorphism function F pq : [0 1] [0 π pq /2] is efine by arcsin pq () = c Bhayo B. A. Sánor J (1 t q ) 1/p t.

2 Inequalities connecting generalize trigonometric functions with Letting t = z 1/q we have arcsin pq () = 1 q q 0 z 1/q 1 (1 z) 1/p z = 1 q B where B(a b ) is incomplete beta function efine as B(a b ) = 0 t a 1 (1 t) b 1 t. ( 1 q 1 1 ) p q The inverse of arcsin pq is enote by sin pq which is efine on the interval [0 π pq /2] where π pq = 2arcsin pq (1) = 2 ( q B 1 q 1 1 ) p 1 = 2 ( 1 q B q 1 1 ) p here B(a b) enote the beta function. We also efine (see [11 Prop. 3.1]) an Letting y = sin pq () we get an arccos pq = arcsin pq ((1 p ) 1/q ) cos pq () = sin pq() [0 π pq /2]. cos pq () = (1 (sin pq ()) q ) 1/p cos pq () p + sin pq () q = 1. (1) The generalize tangent function tan pq () is efine as tan pq () = sin pq() cos pq (). For (0 ) the inverse of generalize hyperbolic sine function sinh pq () is efine by arcsinh pq = 0 (1 + t q ) 1/p t

3 84 B. A. Bhayo J. Sánor an generalize hyperbolic cosine an tangent functions are efine by cosh pq () = sinh pq() tanh pq () = sinh pq() cosh pq () 0 respectively. It follows from the efinitions that cosh pq () p sinh pq () q = 1 0. (2) The main results of the this paper reas as below. Theorem 1. For p q > 1 the following hol 1) For all (0 1) an y (0 π pq /2) with y < arcsin pq () we have arcsin pq () sin pq (y) > y. 2) For all (0 π pq /2) an y (0 1) with tan pq () > y we have tan pq ()arctan pq (y) > y. 3) For all y (0 ) with y < sinh pq () we have sinh pq ()arcsinh pq (y) > y. 4) For all (0 1) an y (0 ) with arctah pq () > y we have Theorem 2. 1) arctah pq ()tanh pq (y) > y. For p q > 1 the following hol arcsin pq () > sin pq(π pq /2) (0 1) π pq /2 2) tan pq() b < arctan pq (b) b = tan pq (k)/k 3) sinh pq() < (0 k) 0 < k < π pq 2 a arctan pq (/a) (0 k) k > 0 a = k sinh pq (k). 4) arctanh pq () > tanh pq(c) c (0 k) k (0 1) c = k/arctanh pq (k).

4 Inequalities connecting generalize trigonometric functions with Preliminaries an proofs The following erivative formulas will be use in our calculations an they can be erive easily from the efinition. Lemma 1. 1) For all (0 π pq /2) we have cos pq() = p q (cos pq()) 2 p (sin pq ()) q 1 2) tan pq() = 1 + p (sin pq ()) q q (cos pq ()) p an for all (0 ) 3) cosh pq() = q p (cosh pq()) 2 p (sinh pq ()) q 1 4) tanh pq() = 1 q (sinh pq ()) q p (cosh pq ()) p. For the following monotone l Hospital rule see [15 Theorem 1.25]. Lemma 2. For < a < b < let f g : [a b] R be continuous on [a b] an be ifferentiable on (a b). Let g () 0 on (a b). If f ()/g () is increasing (ecreasing) on (a b) then so are f() f(a) g() g(a) an f() f(b) g() g(b). If f ()/g () is strictly monotone then the monotonicity in the conclusion is also strict. For the proof of following lemma see ([1]). Lemma 3. Let f : I J be a injective function where I J are the subsets of (0 ). Suppose that the function g() = f()/ I is strictly increasing. Then for any I y J such that f() y following hols f()f 1 (y) y where f 1 : J I enotes the inverse function of f. Uner the same conition if f() y then we have f()f 1 (y) y. (3)

5 86 B. A. Bhayo J. Sánor For the following lemma see [16 Theorem 2 p. 151] [13 Theorem 1]. Lemma 4. 1) Let J R be an open interval an f : J R be a strictly monotonic function. Let f 1 : f(j) J be the inverse of f. If f is concave an increasing then f 1 is conve. 2) For all (0 1) the functions p arcsin p () an p arctanh p () are strictly ecreasing in p (1 ). Lemma 5. For p q > 1 the following hol 1) the function f() = arcsin pq() 2) the function g() = tan pq() is increasing in (0 1) is increasing in (0 π pq /2) 3) the function h() = sinh pq() is increasing in (0 ) 4) the function j() = arctah pq() is increasing in (0 ) with p > q. Proof. Let f() = arcsin pq() = f 1() f 2 (). Then f 1() = (1 q ) 1/p > 0 an f 2() > 0. Now it is clear by Lemma 2 that f is increasing. For the proof of part (2) an (3) let g() = tan pq() = g 1() g 2 () h() = sinh pq() = h 1() h 2 (). Differentiation gives g 1() = 1 + p q (sin pq ()) q (cos pq ()) p > 0 an h 1() = cosh pq () > 0 an the proof is obvious from Lemma 2. For part (4) we get 2 2 tanh pq() = q ( q(sinhpq ()) q 1 (cosh pq ()) p+1 ) q cosh pq () p (sinh pq ()) 2q 1 = = q p (sinh pq()) q 1 (cosh pq ()) 1 2p < 0

6 Inequalities connecting generalize trigonometric functions with since tanh pq () is concave an clearly with p > q it is increasing. By Lemma 4(1) arctah pq () is conve an from this fact we get that arctah pq() is increasing. Hence the rest of proof follows from Lemma 2. Proof of Theorem 1. The functions arcsin pq () tan pq () sinh pq () an arctah pq () are increasing by Lemma 5. The rest of proof follows immeiately from Lemma 3. It is easy to check by using the erivative formulas that the following relations < arcsin pq () (0 1) < tan pq () (0 π pq /2) < sinh pq () (0 ) > tanh pq () arctanh pq () > (0 1). hol true for all p q > 1. By Theorem 1 an above relations we conclue the following corollary. Corollary. 1) 2) 3) 4) For p q > 1 the following hol arcsin pq () < sin pq() (0 1) arctan pq () < tan pq() (0 1) arcsinh pq () < sinh pq() (0 ) arctanh pq () < tanh pq() (0 1).

7 88 B. A. Bhayo J. Sánor Proof of Theorem 2. The monotonicity of the functions imply that Hence arcsin pq () tan pq () sinh pq () f 1 () = π pq 2 arcsin pq() < f 2 () = tan pq() < b f 3 () = a sinh pq () < an f 4 () = carctanh pq () <. arctah pq () f 1 1 () = sin pq(π pq /2) f 1 2 () = arctan pq(b) f 1 3 () = arcsinh pq(/a) f 1 4 () = arctanh pq(c) an the proof follows from (3) if we let y =. Corollary. 1) The following assertions hol true: arcsin() < sin p() for (0 1) p 2 2) sin p() 3) < 2/π p arcsin(2/π p ) for (0 π 2) p (1 2] arctan() < tan p() for (0 1) p (1 2] 4) tan p() < b arctan(b) for (0 k) 0 < k < π p/2 b = tan(k). k The proof follows from Theorem 1 Lemma 4(2) an Corollary 2. Remark. In [17 Theorem 2.3] the following inequalities was prove B(a b ) B(a b y) B(a b + y z) B(a b z) for a (0 1) b > 0 an y > z. Uner the same assumption with 0 < + y z < 1 an y z (0 1) one has arcsin pq ()arcsin pq (y) arcsin pq ( + y z)arcsin pq (z).

8 Inequalities connecting generalize trigonometric functions with References [1] Sánor J. On certain inequalities for hyperbolic an trigonometric functions J. Math. Ineq. [2] Linqvist P. Some remarkable sine an cosine functions. Ricerche i Matematica Vol. XLIV (1995) [3] Bhayo B. A. Vuorinen M. Inequalities for eigenfunctions of the p-laplacian. January pp. arxiv math.ca [4] Bushell P. J. Emuns D. E. Remarks on generalise trigonometric functions. Rocky Mountain J. Math. 42 (2012) Number [5] Biezuner R. J. Ercole G. Martins E. M. Computing the first eigenvalue of the p-laplacian via the inverse power metho. J. Funct. Anal. 257 (2009) no [6] Drábek P. Manásevich R. On the close solution to some p-laplacian nonhomogeneous eigenvalue problems. Differential Integral Equations 12 (1999) no [7] Klén R. Vuorinen M. Zhang X. Inequalities for the generalize trigonometric an hyperbolic functions. October pp. arxiv: [8] Lang J. Emuns D. E. Eigenvalues Embeings an Generalise Trigonometric Functions. Lecture Notes in Mathematics 2016 Springer-Verlag [9] Linqvist P. Peetre J. p-arclength of the q-circle. The Mathematics Stuent Vol. 72 (2003) Nos [10] Bhayo B. A. Vuorinen M. On generalize trigonometric functions with two parameters. J. Appro. Theory 164 (2012) [11] Emuns D. E. Gurka P. Lang J. Properties of generalize trigonometric functions. J. Appro. Theory 164 (2012) oi: /j.jat [12] Takeuchi S. Generalize Jacobian elliptic functions an their application to bifurcation problems associate with p-laplacian. J. Math. Anal. Appl. 385 (2012) oi: /j.jmaa [13] Baricz Á. Bhayo B. A. Vuorinen M. Turán type inequalities for generalize inverse trigonometric functions May pp. arxiv: [math.ca]. [14] Klén R. Visuri M. Vuorinen M. On Joran type inequalities for hyperbolic functions J. Ineq. Appl. vol pp. 14. [15] Anerson G. D. Vamanamurthy M. K. Vuorinen M. Conformal invariants inequalities an quasiconformal maps. J. Wiley pp.

9 90 B. A. Bhayo J. Sánor [16] Kuczma M. An introuction to the theory of functional equations an inequalities. Cauchy s equation an Jensen s inequality. With a Polish summary. Prace Naukowe Uniwersytetu Ślaskiego w Katowicach [Scientific Publications of the University of Silesia] 489. Uniwersytet Ślaski Katowice; Państwowe Wyawnictwo Naukowe (PWN) Warsaw pp. ISBN: [17] Sroysang B. Inequalities for the incomplete beta function Mat. Aeterna 3(2013) no [18] Abramowitz M. an Stegun I. es. Hanbook of mathematical functions with formulas graphs an mathematical tables. National Bureau of Stanars 1964 (Russian translation Nauka 1979). [19] Baricz Á. Functional inequalities involving special functions II. J. Math. Anal. Appl. 327 (2007) no [20] Carlson B. C. Some inequalities for hypergeometric functions. Proc. of Amer. Math. Soc. vol. 17 (1966) no [21] Y.-M Chu Y.-P Jiang an M.-Kun Wang. Inequalities for generalize trigonometric an hyperbolic sine functions. December pp. arxiv: [22] W.-D. Jiang an F. Qi. Geometric conveity of the generalize sine an the generalize hyperbolic sine arxiv The work is receive on September University of Jyväskylä Department of Mathematical Information Technology Jyväskylä Finlan. bhayo.barkat@gmail.com Babeş-Bolyai University Department of Mathematics Str. Kogalniceanu nr Cluj-Napoca Romania. jsanor@math.ubbcluj.ro