MATEMATIQKI VESNIK
originalni nauqni rad
research paper
67, 1 (2015), 17–25
March 2015
POWER MEAN INEQUALITY OF GENERALIZED
TRIGONOMETRIC FUNCTIONS
Barkat Ali Bhayo
Abstract. The author here studies the convexity and concavity properties of the generalized
p-trigonometric functions in the sense of P. Lindqvist with respect to the Power Mean.
1. Introduction
The generalized trigonometric and hyperbolic functions depending on a parameter p > 1 were studied by P. Lindqvist in 1995 [16]. For the case when p = 2,
these functions coincide with elementary functions. Later on numerous authors
have extended this work in various directions, see [8–10, 13, 17]. The generalized
trigonometric function sinp , know as eigenfunction has been a tool in the analysis of more complicated equations, see [6, 7, 11] and the bibliography of these
papers. Here we study the Power Mean inequality of sinp and other generalized
trigonometric functions.
We introduce some notation and terminology for the statement of the main
results.
Given complex numbers a, b and c with c 6= 0, −1, −2, . . . , the Gaussian hypergeometric function is the analytic continuation to the slit place C \ [1, ∞) of the
series
∞
X
(a, n)(b, n) z n
F (a, b; c; z) = 2 F1 (a, b; c; z) =
,
|z| < 1.
(c, n)
n!
n=0
Here (a, 0) = 1 for a 6= 0, and (a, n) is the shifted factorial function or the Appell
symbol
(a, n) = a(a + 1)(a + 2) · · · (a + n − 1)
for n ∈ Z+ , see [1]. The integral representation of the hypergeometric function is
given as follows [21, p. 20]
Z 1
Γ(c)
F (a, b; c; z) =
tb−1 (1 − t)c−b−1 (1 − zt)−a dt
(1.1)
Γ(c)(c − b) 0
Re(c) > Re(a) > 0, | arg(1 − z)| < π.
2010 Mathematics Subject Classification: 33C99, 33B99
Keywords and phrases: Eigenfunctions sinp ; Power Mean; generalized trigonometric functions.
17
18
Power mean inequality
Let us start the discussion of eigenfunctions of one-dimensional p-Laplacian
∆p on (0, 1), p ∈ (1, ∞). The eigenvalue problem [13]
¡
¢0
−∆p u = − |u0 |p−2 u0 = λ|u|p−2 u, u(0) = u(1) = 0,
has eigenvalues λn = (p − 1)(nπp )p , and eigenfunctions sinp (nπp t), n ∈ N, where
sinp is the inverse function of arcsinp , which is defined below, and
µ
¶
Z
2 1
2
1 1
2π
πp =
(1 − s)−1/p s1/p−1 ds = B 1 − ,
=
,
p 0
p
p p
p sin(π/p)
with π2 = π.
Let us consider the following homeomorphisms
sinp : (0, ap ) → I,
cosp : (0, ap ) → I,
sinhp : (0, ∞) → I,
tanhp : (0, ∞) → I,
where I = (0, 1) and
ap =
tanp : (0, bp ) → I,
πp
, bp = 2−1/p F
2
µ
1 1
1 1
, ;1 + ;
p p
p 2
¶
.
From integral formula (1.1) and by using the change of variables we define the
inverse functions of the above homeomorphisms for x ∈ I,
µ
¶
Z x
1 1
1
arcsinp x =
(1 − tp )−1/p dt = x F
, ; 1 + ; xp
p p
p
0
¶
µ
1
= x(1 − xp )(p−1)/p F 1, 1; 1 + ; xp ,
p
µ
¶
Z x
1
1
arctanp x =
(1 + tp )−1 dt = x F 1, ; 1 + ; −xp
p
p
0
µ
¶1/p µ
¶
p
p
x
1 1
1
x
=
F
, ;1 + ;
,
1 + xp
p p
p 1 + xp
µ
¶
Z x
1 1
1
p −1/p
p
arsinhp x =
(1 + t )
dt = x F
, ; 1 + ; −x
p p
p
0
µ
¶
µ
¶
1/p
xp
1
1
xp
=
F 1, ; 1 + ;
,
p
1+x
p
p 1 + xp
¶
µ
Z x
1
1
artanhp x =
(1 − tp )−1 dt = x F 1 , ; 1 + ; xp ,
p
p
0
and by [10, Prop. 2.2] arccosp x = arcsinp ((1 − xp )1/p ). The special case of above
functions for p = 2 is defined in terms of hypergeometric functions in [2, p. 8]. In
particular, these functions reduce to the familiar functions for the case p = 2.
For t ∈ R and x, y > 0, the Power Mean Mt of order t is defined by
µ t
¶1/t
 x + yt
, t 6= 0,
Mt =
2
√
x y,
t = 0.
19
B.A. Bhayo
The main results of the paper are the following theorems:
Theorem 1.1. For p > 1, t ≥ 0 and r, s ∈ (0, 1), we have
(1) arcsinp (Mt (r, s)) ≤ Mt (arcsinp (r), arcsinp (s)),
(2) artanhp (Mt (r, s)) ≤ Mt (artanhp (r), artanhp (s)),
(3) arctanp (Mt (r, s)) ≥ Mt (arctanp (r), arctanp (s)),
(4) arsinhp (Mt (r, s)) ≥ Mt (arsinhp (r), arsinhp (s)).
Theorem 1.2. For p > 1, t ≥ 1 and r, s ∈ (0, 1), the following relations hold
(1) sinp (Mt (r, s)) ≥ Mt (sinp (r), sinp (s)),
(2) cosp (Mt (r, s)) ≤ Mt (cosp (r), cosp (s)),
(3) tanp (Mt (r, s)) ≤ Mt (tanp (r), tanp (s)),
(4) tanhp (Mt (r, s)) ≥ Mt (tanhp (r), tanhp (s)),
(5) sinhp (Mt (r, s)) ≤ Mt (sinhp (r), arsinhp (s)).
The above results also generalize some results of [4] (Theorem 2.5, Lemma
2.9), which are the special cases of the above theorems when t = 0 and t = 2.
Generalized convexity/concavity with respect to general mean values has been
studied recently in [3].
Let f : I → (0, ∞) be continuous, where I is a subinterval of (0, ∞). Let Mt
be a Power Mean. We say that f is Mt Mt -convex (concave) if
f (Mt (x, y)) ≤ (≥)Mt (f (x), f (y)) for all x, y ∈ I.
In conclusion, we see that the above results are (Mt , Mt )-convexity or (Mt , Mt )concavity properties of the functions involved. In view of [3], it is natural to expect
that similar results might also hold for some other pairs (M, N ) of mean values.
2. Preliminaries and proofs
For easy reference we record the following lemma from [2], which is sometimes
called the monotone l’Hospital rule.
Lemma 2.1. [2, Theorem 1.25] For −∞ < a < b < ∞, let f, g : [a, b] → R
be continuous on [a, b], and be differentiable on (a, b). Let g 0 (x) 6= 0 on (a, b). If
f 0 (x)/g 0 (x) is increasing (decreasing) on (a, b), then so are
[f (x) − f (a)]/[g(x) − g(a)]
and
[f (x) − f (b)]/[g(x) − g(b)].
If f 0 (x)/g 0 (x) is strictly monotone, then the monotonicity in the conclusion is also
strict.
For the next two lemmas see [4, Theorems 1.1, 1.2, 2.5 & Lemma 3.6].
Lemma 2.2. For p > 1 and x ∈ (0, 1), we have
20
Power mean inequality
µ
¶
xp
πp
(1) 1 +
x < arcsinp x <
x,
p(1 + p)
2
µ
¶
1 − xp
πp
(2) 1 +
(1 − xp )1/p < arccosp x <
(1 − xp )1/p ,
p(1 + p)
2
µ
¶1/p
(p(1 + p)(1 + xp ) + xp )x
xp
1/p
(3)
< arctanp x < 2
bp
,
1 + xp
p(1 + p)(1 + xp )1+1/p
µ
¶
µ
¶1/p
³
´
log(1 + xp )
xp
(4) z 1 +
< arsinhp x < z 1 + p1 log(1 + xp ) , z =
,
1+p
1 + xp
µ
¶
´
³
1
p
(5) x 1 −
log(1 − x ) < artanhp x < x 1 − p1 log(1 − xp ) .
1+p
(1)
(2)
(3)
(4)
(5)
Lemma 2.3. For p, q > 1 and r, s ∈ (0, 1), the following inequalities hold:
p
√
arcsinp ( r s) ≤ arcsinp (r) arcsinp (s),
p
√
artanhp ( r s) ≤ artanhp (r) artanhp (s),
p
√
arsinhp (r) arsinhp (s) ≤ arsinhp ( r s),
p
√
arctanp (r) arctanp (s) ≤ arctanp ( r s),
√
π√p q ≤ πp πq .
Lemma 2.4. For m ≥ −1, p > 1, the following functions
¶m
µ
d
arcsinp x
(arcsinp x),
(1) f1 (x) =
x
dx
µ
¶m
artanhp x
d
(2) f2 (x) =
(artanhp x)
x
dx
are increasing in x ∈ (0, 1), and
µ
¶m
arctanp x
d
(3) f3 (x) =
(arctanp x),
x
dx
¶m
µ
d
arsinhp x
(arsinhp x)
(4) f4 (x) =
x
dx
are decreasing in x ∈ (0, 1).
Proof. By definition,
µ
f1 (x) =
arcsinp x
x
¶m
1
.
(1 − xp )1/p
´m
³
arcsinp x
is increasing by Lemma 2.1, and clearly (1 − xp )1/p is
For m ≥ 0,
x
increasing. For the case m ∈ [−1, 0), we define
µ
¶s
x
1
, s ∈ (0, 1].
h1 (x) =
arcsinp x
(1 − xp )1/p
21
B.A. Bhayo
We get
ξ
((1 − xp )1/p (xp + s(1 − xp ))F1 (x) − s(1 − xp ))
1 − xp
µ
¶
ξ
xp
p
p 1/p p
p
>
(1 − x ) (x + s(1 − x ))(1 +
) − s(1 − x ) > 0,
1 − xp
p(1 + p)
h01 (x) =
by Lemma 2.2(1), where
ξ=
(1 − xp )−(1+2/p)
x
µ
1
F1 (x)
¶1+s
µ
and
F1 (x) = F
¶
1 1
1
, ; 1 + ; xp .
p p
p
For (2), clearly f2 is increasing for m ≥ 0. For the case when m ∈ [−1, 0), we
define
µ
¶s
x
1
h2 (x) =
, s ∈ (0, 1].
artanhp x)
1 − xp
Differentiating with respect to x, we get
h02 (x) =
(F2 (x))
−(1+s)
((pxp − sxp + s) F2 (x) − s)
2
³
´
where F2 (x) = F 1, p1 ; 1 + p1 ; xp .
x (xp − 1)
> 0,
For (3), the proof for the case when m ≥ 0 follows similarly from Lemma 2.1.
For the case m ∈ [−1, 0), let
µ
¶−s
arctanp x
d
h3 (x) =
(arctanp x), s ∈ (0, 1].
x
dx
We have
h03 (x) =
F3 (x)−(1+s)
((s + s rp − p rp )F3 (x) − s)
r(1 + rp )2
<
F3 (x)−(1+s)
((s + s rp − s rp )F3 (x) − s)
r(1 + rp )2
s F3 (x)−(1+s)
(1 − F3 (x)) < 0,
r(1 + rp )2
³
´
where F3 (x) = F 1, p1 ; 1 + p1 ; −xp
=−
For (4), when m ≥ 0, the proof follows from Lemma 2.1. For m ∈ [−1, 0), let
µ
¶s
x
1
h4 (x) =
, s ∈ (0, 1].
arsinhp x
(1 + xp )1/p
We have
h04 (x) = γ((1 − xp )1/p (s(1 − xp ) − xp )F4 (x) − s(1 − xp ))
³
³
³
´
´´
1
1
< γ s(1 + xp ) 1 + log(1 + xp ) − s(1 + xp ) − xp 1 +
log(1 + xp )
p
1+p
γ
p
p
p
(s(1 + x )(1 + p) log(1 + x ) − p(1 + p)x − p xp log(1 + xp )) < 0,
=
p(1 + p)
22
Power mean inequality
by Lemma 2.2(4), where
µ
¶1+s
(1 − xp )−(1+2/p)
1
γ=
x
F4 (x)
µ
and F4 (x) = F
¶
1 1
1
p
, ; 1 + ; −x .
p p
p
Proof of Theorem 1.1. Let 0 < x < y < 1, and u = ((xt + y t )/2)1/t > x. We
denote arcsin(x), artanh(x), arctan(x), arsinh(x) by gi (x), i = 1, . . . , 4 respectively,
and define
gi (x)t + gi (y)t
g(x) = gi (u)t −
.
2
Differentiating with respect to x, we get du/dx = (1/2)(x/u)t−1 and
³ x ´t−1 1
1
d
d
g 0 (x) = t gi (x)t−1 (gi (u))
− t gi (x)t−1 (gi (x))
2
du
u
2
dx
t
= xt−1 (fi (u) − fi (x)),
2
where
µ
fi (x) =
gi (x)
x
¶t−1
d
(gi (x)), i = 1, . . . , 4.
dx
By Lemma 2.4, g 0 is positive and negative for fi=1,2 and fi=3,4 , respectively. This
implies that
g(x) < (>)g(y) = 0,
for gi=1,2 and gi=3,4 , respectively. The case when t = 0 follows from Lemma 2.3.
This completes the proof.
Lemma 2.5. For p > 1 and s ∈ (0, 1), the function
³
³
´´
³ π ´−s p − π cot π/p csc(π/p)
p
f (p) =
p
p3
is decreasing in p ∈ (1, ∞).
Proof. We have
·
µ ¶
µ ¶
µ ¶¸
π
π
π
0
2
2
2
2
2
f (p) = ξ 2p (1 − s) + π (1 − s) cot
− πp(4 − 3s) cot
+ π csc
,
p
p
p
where
(2π)−s
ξ=−
p3
µ
csc(π/2)
p2
¶1−s
,
which is negative.
Lemma 2.6. [14, Thm 2, p.151] Let J ⊂ R be an open interval, and let
f : J → R be strictly monotonic function. Let f −1 : f (J) → J be the inverse to f
then
B.A. Bhayo
(1)
(2)
(3)
(4)
if
if
if
if
f
f
f
f
is
is
is
is
23
convex and increasing, then f −1 is concave,
convex and decreasing, then f −1 is convex,
concave and increasing, then f −1 is convex,
concave and decreasing, then f −1 is concave.
Lemma 2.7. For m ≥ 1, p > 1 and x ∈ (0, 1), the following functions
µ
¶m−1
d
sinp x
(1) h1 (x) =
(sinp x),
x
dx
µ
¶m−1
tanhp x
d
(tanhp x),
(2) h2 (x) =
x
dx
are decreasing in x, and
³ cos x ´m−1 d
p
(3) h3 (x) =
(cosp x),
x
dx
µ
¶m−1
tanp x
d
(4) h4 (x) =
(tanp x),
x
dx
µ
¶m−1
sinhp x
d
(5) h5 (x) =
(sinhp x),
x
dx
are increasing in x.
Proof. Let f (x) = arcsinp x, x ∈ (0, 1). We get
1
f 0 (x) =
,
(1 − xp )1/p
which is positive and increasing, hence f is convex. Clearly sinp x is increasing, and
d
sinp x is decreasing, and (sinp x)/x is
by Lemma 2.6 is concave, this implies that dx
d
decreasing also by Lemma 2.1. Similarly we get that dx
tanhp x is decreasing and
d
d
d
dx cosp x, dx tanp x, dx sinhp x are increasing, and the rest of proof follows from
Lemma 2.1.
Proof of Theorem 1.2. The proof is similar to the proof of Theorem 1.1 and
follows from Lemma 2.7.
Proposition 2.8. For p, q > 1 and t < 1, we have
πMt (p,q) ≤ Mt (πp , πq ).
Proof. Let 1 < p < q < ∞, and w = ((pt + q t )/2)1/t > p. We define
(πp )t + (πq )t
.
2
Differentiating with respect to p, we get dw/dp = (1/2)(p/w)t−1 and
³ p ´t−1 1
d
d
1
− t (πp )t−1 (πp )
g 0 (p) = t (πp )t−1 (πw )
2
dx
w
2
dx
t t−1
= p (f (w) − f (p)),
2
g(p) = (πp )t −
24
Power mean inequality
where
µ
f (p) =
πp
p
¶t−1
d
πp .
dp
Clearly πp is decreasing, hence (πp /p)t−1 is increasing for t < 1 and d/dp(πp )
is increasing by the proof of Lemma [4, Lemma 3.6]. This implies that f (p) is
increasing, and it follows that g is increasing. Hence g(p) < g(q) = 0. The case
when t = 0 follows from Lemma 2.3(5). This completes the proof.
The following corollary follows immediately from Lemma 2.7.
(1)
(2)
(3)
(4)
(5)
Corollary 2.9. For p > 1 and r, s ∈ (0, 1) with r ≤ s, we have
sinp r
sinp s
≥
,
r
s
cosp r
cosp s
≥
,
r
s
tanp r
tanp s
≤
,
r
s
sinhp r
sinhp s
≤
,
r
s
tanhp r
tanhp s
≥
.
r
s
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(received 21.04.2013; in revised form 13.10.2013; available online 15.11.2013)
Department of Mathematical Information Technology, University of Jyväskylä, FI-40014 Jyväskylä,
Finland
E-mail: bhayo.barkat@gmail.com