Andreev-Korkin Identity, Saigo Fractional Integration Operator and LipL(α) Functions
The Andreev-Korkin identity for the Chebyshev functional is treated by Holder inequality, when the functional consists of LipL(α) functions. The derived upper bound is applied to the so-called Chebyshev-Saigo functional, built by Saigo fractional integral operator - recently introduced by Saxena et...
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irk-123456789-1067152016-10-04T03:02:21Z Andreev-Korkin Identity, Saigo Fractional Integration Operator and LipL(α) Functions Jankov, D. Pogány, T.K. The Andreev-Korkin identity for the Chebyshev functional is treated by Holder inequality, when the functional consists of LipL(α) functions. The derived upper bound is applied to the so-called Chebyshev-Saigo functional, built by Saigo fractional integral operator - recently introduced by Saxena et al. (R.K. Saxena, J. Ram, J. Daiya, and T.K. Pogany - Integral Tranforms Spec. Funct. 22 (2011), 671-680). К тождеству Андреева-Коркина для функционала Чебышева с функциями применяется неравенство Гёльдера. Полученная верхняя граница применяется к так называемому функционалу Чебышева-Сеге, построенному при помощи оператора Сеге дробного интегрирования, предложенного недавно Р.К. Саксеной и др. (R.K. Saxena, J. Ram, J. Daiya, and T.K. Pogány. - Integral Tranforms Spec. Funct. 22 (2011), 671-680). 2012 Article Andreev-Korkin Identity, Saigo Fractional Integration Operator and LipL(α) Functions / D. Jankov, T.K. Pogány // Журнал математической физики, анализа, геометрии. — 2012. — Т. 8, № 2. — С. 144-157. — Бібліогр.: 9 назв. — англ. 1812-9471 http://dspace.nbuv.gov.ua/handle/123456789/106715 en Журнал математической физики, анализа, геометрии Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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The Andreev-Korkin identity for the Chebyshev functional is treated by Holder inequality, when the functional consists of LipL(α) functions. The derived upper bound is applied to the so-called Chebyshev-Saigo functional, built by Saigo fractional integral operator - recently introduced by Saxena et al. (R.K. Saxena, J. Ram, J. Daiya, and T.K. Pogany - Integral Tranforms Spec. Funct. 22 (2011), 671-680). |
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Jankov, D. Pogány, T.K. Andreev-Korkin Identity, Saigo Fractional Integration Operator and LipL(α) Functions Журнал математической физики, анализа, геометрии |
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Jankov, D. Pogány, T.K. |
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Jankov, D. |
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Andreev-Korkin Identity, Saigo Fractional Integration Operator and LipL(α) Functions |
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Andreev-Korkin Identity, Saigo Fractional Integration Operator and LipL(α) Functions |
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Andreev-Korkin Identity, Saigo Fractional Integration Operator and LipL(α) Functions |
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Andreev-Korkin Identity, Saigo Fractional Integration Operator and LipL(α) Functions |
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Andreev-Korkin Identity, Saigo Fractional Integration Operator and LipL(α) Functions |
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andreev-korkin identity, saigo fractional integration operator and lipl(α) functions |
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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Andreev-Korkin Identity, Saigo Fractional Integration Operator and LipL(α) Functions / D. Jankov, T.K. Pogány // Журнал математической физики, анализа, геометрии. — 2012. — Т. 8, № 2. — С. 144-157. — Бібліогр.: 9 назв. — англ. |
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Журнал математической физики, анализа, геометрии |
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AT jankovd andreevkorkinidentitysaigofractionalintegrationoperatorandliplafunctions AT poganytk andreevkorkinidentitysaigofractionalintegrationoperatorandliplafunctions |
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Journal of Mathematical Physics, Analysis, Geometry
2012, vol. 8, No. 2, pp. 144�157
Andreev�Korkin Identity, Saigo Fractional Integration
Operator and LipL(α) Functions
D. Jankov
Department of Mathematics, University of Osijek
Trg Lj. Gaja 6, 31000 Osijek, Croatia
E-mail: djankov@mathos.hr
T.K. Pog�any
Faculty od Maritime Studies, University of Rijeka
Studentska 2, 51000 Rijeka, Croatia
E-mail: poganj@pfri.hr
Received October 26, 2010, revised May 25, 2011
The Andreev�Korkin identity for the Chebyshev functional is treated by
H�older inequality, when the functional consists of LipL(α) functions. The
derived upper bound is applied to the so-called Chebyshev�Saigo functional,
built by Saigo fractional integral operator � recently introduced by Saxena
et al. (R.K. Saxena, J. Ram, J. Daiya, and T.K. Pog�any. � Integral
Tranforms Spec. Funct. 22 (2011), 671�680).
Key words: Chebyshev functional, Andreev�Korkin identity, Chebyshev�
Saigo functional, Saigo hypergeometric fractional integration operator, Lip-
schitz function clas.
Mathematics Subject Classi�cation 2000: 26D15, 26A16 (primary); 26A33,
26D10 (secondary).
1. Introduction
Let w : [0, 1] 7→ R+, w ∈ L1[0, 1] be a normalized weight function, that is,
1∫
0
w(x)dx = 1 .
The weighted Chebyshev functional is de�ned by
T(w; f, g) := M(w; fg)−M(w; f)M(w; g) , (1.1)
c© D. Jankov and T.K. Pog�any, 2012
Andreev�Korkin Identity, Saigo Fractional Integration Operator
where M(w; f) denotes the integral mean
M(w; f) =
1∫
0
w(x)f(x)dx . (1.2)
We point out that using another support interval supp(f) = [a, b] ⊂ R, say,
di�erent from the unit one, we only achieve an arti�cial extension of (1.3), since
obvious substitution
x− a
b− a
: [a, b] 7→ [0, 1]
leads us to (1.1).
Now let us recall in short the Andreev�Korkin identity for the weighted Cheby-
shev functional. In our setting this celebrated relation reads as follows:
T(w; f, g) =
1
2
1∫
0
1∫
0
w(x)w(y)(f(x)− f(y))(g(x)− g(y)) dxdy . (1.3)
R e m a r k 1.1. According to [1, pp. 6�7] starting from the �nite sums identity
1
n
n∑
j=1
xjyj =
( 1
n
n∑
j=1
xj
)( 1
n
n∑
j=1
yj
)
+
1
n2
∑
1≤i<j≤n
(xi − xj)(yi − yj) , (1.4)
Korkin proved in 1882 Chebyshev's integral inequality [1, p. 2, Eq. (0.3)] in his
letter to Bugaev [2] (in Russian) and presented the same procedure in the letter
to Hermite [3] (in French), see also the most familiar source [4, pp. 242�243].
The identity analogous to Korkin's (1.4), where integrals replaced �nite sums,
was obtained in the next year by Andreev [5], another mathematician from the
celebrated Kharkiv Mathematical Society.
It seems that A. Winckler (1884) and F. Franklin (1885) rediscovered inde-
pendently Korkin's and Andreev's identities, respectively [1, p. 8], but we prefer
to call (1.3) the Andreev�Korkin identity.
2. Andreev�Korkin Identity Built in Lipschitz Function Class
Given two metric spaces (Ξ, d) and (Υ, d), where d(x, y) = |x − y|, x, y ∈
Ξ, Υ ⊆ R. A function f : Ξ 7→ Υ is said to be uniform Lipschitz of order α on Ξ
if there exists an absolute constant L > 0 such that
|f(x)− f(y)| ≤ L|x− y|α 0 < α ≤ 1, x, y ∈ Ξ . (2.1)
Journal of Mathematical Physics, Analysis, Geometry, 2012, vol. 8, No. 2 145
D. Jankov and T.K. Pog�any
Here L is the Lipschitz constant, and the class consisting of such functions we
write LipL(α).
Theorem 2.1. Let r, s, r−1 +s−1 = 1, r > 1, be conjugated H�older exponents.
Assume that f ∈ LipLf
(αf ), f ∈ LipLg
(αg). Then
∣∣T(w; f, g)
∣∣ ≤ LfLg
2
min
{
M1,M2
}
, (2.2)
where
M1 =
( 1∫
0
1∫
0
w(x)w(y)|x− y|αf rdxdy
)1/r( 1∫
0
1∫
0
w(x)w(y)|x− y|αgsdxdy
)1/s
,
(2.3)
M2 =
1
(
αfr + 1
)1/r(
αgs + 1)1/s
( 1∫
0
xαf r+1
{
wr(x) + wr(1− x)
}
dx
)1/r
×
( 1∫
0
xαgs+1
{
ws(x) + ws(1− x)
}
dx
)1/s
. (2.4)
P r o o f. By the triangle inequality and since r, s, r > 1 are conjugated, we
conclude by virtue of the weighted H�older inequality from the Andreev�Korkin
identity the following estimates:
∣∣T(w; f, g)
∣∣ ≤ 1
2
1∫
0
1∫
0
{
w(x)w(y)
}1/r+1/s∣∣f(x)− f(y)
∣∣ ∣∣g(x)− g(y)
∣∣ dxdy
≤ 1
2
( 1∫
0
1∫
0
w(x)w(y)
∣∣f(x)− f(y)
∣∣r dxdy
)1/r
×
( 1∫
0
1∫
0
w(x)w(y)
∣∣g(x)− g(y)
∣∣s dxdy
)1/s
. (2.5)
Because f ∈ LipLf
(αf ), g ∈ LipLg
(αg), we estimate the right�hand side of (2.5)
by
146 Journal of Mathematical Physics, Analysis, Geometry, 2012, vol. 8, No. 2
Andreev�Korkin Identity, Saigo Fractional Integration Operator
∣∣T(w; f, g)
∣∣ ≤ LfLg
2
( 1∫
0
1∫
0
w(x)w(y)|x− y|αf r dxdy
)1/r
×
( 1∫
0
1∫
0
w(x)w(y)|x− y|αgs dxdy
)1/s
which is evidently (2.3) up to the constant.
To prove (2.4), we begin with regrouping the integrand in (2.5) separating two
weight functions and employ the classical H�older inequality with the same couple
of conjugated parameters r, s getting
∣∣T(w; f, g)
∣∣ ≤ 1
2
1∫
0
1∫
0
{
w(x)
∣∣f(x)− f(y)
∣∣} · {w(y)
∣∣g(x)− g(y)
∣∣} dxdy
≤ 1
2
( 1∫
0
1∫
0
wr(x)
∣∣f(x)− f(y)
∣∣r dxdy
)1/r
×
( 1∫
0
1∫
0
ws(y)
∣∣g(x)− g(y)
∣∣s dxdy
)1/s
.
Estimating the increments of f and g by their LipL(α) de�nition, we conclude
∣∣T(w; f, g)
∣∣ ≤ LfLg
2
( 1∫
0
1∫
0
wr(x)|x− y|αf r dxdy
)1/r
×
( 1∫
0
1∫
0
ws(y)|x− y|αgs dxdy
)1/s
. (2.6)
Since
1∫
0
|x− y|αf rdy =
x∫
0
(x− y)αf rdy +
1∫
x
(y − x)αf rdy
=
1
αfr + 1
(
xαf r+1 + (1− x)αf r+1
)
,
the substitution of arguments leads us via (2.6) to the stated upper bound (2.4).
Journal of Mathematical Physics, Analysis, Geometry, 2012, vol. 8, No. 2 147
D. Jankov and T.K. Pog�any
Now we will apply the result obtained to the weight function case closely
connected to the Chebyshev�Saigo functional associated with the Saigo fractional
integral operator.
3. Andreev�Korkin Identity for Chebyshev�Saigo Functional
The Saigo hypergeometric fractional integral of the function f : R+ 7→ R is
de�ned for all η > 0, σ ∈ R as
Iρ,σ,η
0,t
[
f
]
=
tσ
Γ(ρ)
1∫
0
(1− x)ρ−1
2F1
[ ρ− σ,−η
ρ
∣∣∣1− x
]
f(tx)dx <(ρ) > 0
dn
dtn
Iρ+n,−σ−n,η−n
0,t
[
f
] <(ρ) ≤ 0, n =
[<(−ρ)
]
+ 1 ,
(3.1)
where Γ(·) stands for the Euler gamma function, compare, for instance, [6, p. 104,
De�nition 3.20].
The Riemann�Liouville and Erd�elyi�Kober fractional integration operators
follow respectively as special cases of (3.1), viz.
Iρ,ρ,η
0,t
[
f
]
= I ρ
0,t
[
f
]
=
tρ
Γ(ρ)
1∫
0
(1− x)ρ−1 f(tx) dx <(ρ) > 0, (3.2)
Iρ,0,η
0,t
[
f
]
= Iρ,η
0,t
[
f
]
=
1
Γ(ρ)
1∫
0
(1− x)ρ−1xη f(tx) dx <(ρ), η > 0 . (3.3)
The hypergeometric term in the Saigo operator's integrand is strictly positive [7,
p. 35, Theorem 2, Eqs. (3.3), (3.4)]
2F1
[ ρ− σ, −η
ρ
∣∣∣x
]
> 0, x ∈ (0, 1) .
Hence, for all σ > −1, the related weight function
wS(x) =
Γ(1 + σ)Γ(1 + ρ + η)
Γ(ρ)Γ(1 + σ + η)
(1− x)ρ−1
2F1
[ ρ− σ,−η
ρ
∣∣∣1− x
]
(3.4)
is well-de�ned. Moreover, the associated weight functions, relative to the Riemann�
Liouville and the Erd�elyi�Kober operators, are
wRL(x) = ρ (1− x)ρ−1 ρ > 0, (3.5)
wEK(x) =
(1− x)ρ−1xη
B(ρ, 1 + η)
min{ρ, 1 + η} > 0, (3.6)
148 Journal of Mathematical Physics, Analysis, Geometry, 2012, vol. 8, No. 2
Andreev�Korkin Identity, Saigo Fractional Integration Operator
respectively. Display (3.5) is obvious, while by virtue of the fact
2F1
[ ρ, −η
ρ
∣∣∣ ·
]
= 1F0
[ −η
−
∣∣∣ ·
]
= (1− ·)η ,
we deduce (3.6). (We point out that all three considered weight functions are
independent of any scaling parameter t). Now we are ready to introduce the
scaled integral mean associated with the Saigo fractional integral operator in the
form
Mt(wS ; f) :=
1∫
0
wS(x)f(tx) dx t > 0. (3.7)
Of course t = 1, that is, M1 ≡ M gives a link to the integral mean (1.2).
De�nition 3.1. The Chebyshev weighted scaled functionals
TS(f, g) := Mt(wS ; fg)−Mt(wS ; f)Mt(wS ; g), (3.8)
TRL(f, g) := Mt(wRL; fg)−Mt(wRL; f)Mt(wRL; g), (3.9)
TEK(f, g) := Mt(wEK ; fg)−Mt(wEK ; f)Mt(wEK ; g) (3.10)
we call, by convention, the Chebyshev�Saigo, the Riemann�Liouville and the Erd�elyi�
Kober functionals, respectively, where Mt(w; ·) is given by (3.7).
R e m a r k 2.1. The Chebyshev�Saigo functional was introduced in a some-
what di�erent manner by Saxena et al. in [8, Eq. (2.8)].
Theorem 3.1. Let r, s, r−1+s−1 = 1, r > 1, be conjugated H�older parameters.
Then for all f ∈ LipLf
(αf ), g ∈ LipLg
(αg) and min{t,<(ρ), η} > 0, σ ∈ R, we
have
∣∣TS(f, g)
∣∣ ≤ LfLg Γ2(1 + σ) Γ2(1 + ρ + η) E1/r(r, αf ) E1/s(s, αg)
2
(
αfr + 1
)1/r(
αgs + 1
)1/s Γ2(ρ) Γ2(1 + σ + η)
tαf+αg , (3.11)
where
E(u, v) =
1∫
0
xuv+1
{
(1− x)u(ρ−1)
2F
r
1
[ ρ− σ, −η
ρ
∣∣∣ 1− x
]
+ xu(ρ−1)
2F
r
1
[ ρ− σ, −η
ρ
∣∣∣x
]}
dx .
P r o o f. A straightforward application of Theorem 2.1 results in (3.11).
Indeed, following the lines of the proving procedure of Theorem 2.1 we have
∣∣TS(f, g)
∣∣ ≤ 1
2
1∫
0
1∫
0
{
wS(x)
∣∣f(tx)− f(ty)
∣∣} · {wS(y)
∣∣g(tx)− g(ty)
∣∣} dxdy
Journal of Mathematical Physics, Analysis, Geometry, 2012, vol. 8, No. 2 149
D. Jankov and T.K. Pog�any
≤ 1
2
( 1∫
0
1∫
0
wr
S(x)
∣∣f(tx)− f(ty)
∣∣r dxdy
)1/r
×
( 1∫
0
1∫
0
ws
S(y)
∣∣g(tx)− g(ty)
∣∣s dxdy
)1/s
=: U .
Both f and g being Lipschitz, we may conclude
U ≤ LfLg
2
( 1∫
0
1∫
0
wr
S(x)|x− y|αf r dxdy
)1/r
×
( 1∫
0
1∫
0
ws
S(y)|x− y|αgs dxdy
)1/s
· tαf+αg .
Now obvious further calculation leads to (3.11).
The next results show how to reduce upper bounds for the modulus of the
Chebyshev�Saigo functional to the bounds when the Saigo hypergeometric frac-
tional integration operator is replaced by the Riemann�Liouville and the Erd�elyi�
Kober operators.
Corollary 3.1. Let r, s, r−1 + s−1 = 1, r > 1, be conjugated H�older para-
meters. Then for all f ∈ LipLf
(αf ), g ∈ LipLg
(αg) and min{t,<(ρ)} > 0, we
have ∣∣TRL(f, g)
∣∣ ≤ LfLg ρ2 G1/r(r, αf )G1/s(s, αf )
2
(
αfr + 1
)1/r(
αgs + 1
)1/s
tαf+αg , (3.12)
where
G(u, v) = B(uv + 2, u(ρ− 1) + 1) +
(
(v + ρ− 1)u + 2
)−1
.
Corollary 3.2. Let r, s, r−1+s−1 = 1, r > 1, be conjugated H�older exponents.
Then for all f ∈ LipLf
(αf ), g ∈ LipLg
(αg) and min{t,<(ρ), η} > 0, we have
∣∣TEK(f, g)
∣∣ ≤ LfLg H1/r(r, αf )H1/s(s, αg)
2
(
αfr + 1
)1/r(
αgs + 1
)1/s B2(ρ, 1 + η)
tαf+αg , (3.13)
where
H(u, v) = B
(
(v + η)u + 2, u(ρ− 1) + 1
)
+ B
(
u(v + ρ− 1) + 2, ηu + 1
)
.
150 Journal of Mathematical Physics, Analysis, Geometry, 2012, vol. 8, No. 2
Andreev�Korkin Identity, Saigo Fractional Integration Operator
4. Further Bounds for the Chebyshev�Saigo Functional
To get a more sophisticated bound for TS(f, g, ), we need an auxiliary upper
bound inequality for the hypergeometric function appearing in the Saigo fractional
integral operator. A similar upper bound was given by Carlson [7, p. 35, Theorem
2, Eqs. (2.13), (2.14)]:
Lemma. Let c > b > 0 and x < 1, x 6= 0. Then
2F1
[ a, b
c
∣∣∣x
]
<
{
J a < −1,
min
{
H, J1
}
a ∈ (−1, 0),
(4.1)
where
J := (1− b/c) + (b/c)(1− x)−a,
H :=
(
1− bx/c
)−a
,
J1 := (b/c)(1− x)c−a−b + (1− b/c)(1− x)−b .
If a ≤ c− 1, then min{H, J1} = H.
In order to present the results, we need a de�nition of the Fox�Wright function
pΨ∗
q which is a generalization of the familiar generalized hypergeometric function
pFq [6, 9],
pΨ∗
q
[ (a1, A1), · · · , (ap, Ap)
(b1, B1), · · · , (bq, Bq)
∣∣∣ z
]
= pΨ∗
q
[ (ap, Ap)
(bq, Bq)
∣∣∣ z
]
:=
∞∑
n=0
∏p
j=1(aj)Ajn∏q
j=1(bj)Bjn
zn
n!
, (4.2)
where (τ)T is the Pochhammer symbol (or shifted factorial), with (1)n = n!, n ∈
N0, de�ned in terms of gamma function by
(τ)T =
Γ(τ + T )
Γ(τ)
=
{
1 T = 0, τ ∈ C \ {0},
τ(τ + 1) · · · (τ + T − 1) T ∈ N, τ ∈ C,
where, as understood conventionally, (0)0 := 1.
In (4.2) aj , bk ∈ C, Aj , Bk > 0, j = 1, p, k = 1, q and
∆ = 1 +
q∑
j=1
Bj −
p∑
j=1
Aj ≥ 0 ; (4.3)
for ∆ = 0 the convergence holds for suitably bounded values of |z|, given by
|z| < ∇, where
∇ =
q∏
j=1
B
Bj
j ·
p∏
j=1
A
−Aj
j . (4.4)
Journal of Mathematical Physics, Analysis, Geometry, 2012, vol. 8, No. 2 151
D. Jankov and T.K. Pog�any
Theorem 4.1. Let r, s, r−1+s−1 = 1, r > 1, be conjugated H�older parameters.
Then for all f ∈ LipLf
(αf ), g ∈ LipLg
(αg), 0 < σ < ρ < 2σ we have:
(i) for η > 1,
∣∣TS(f, g)
∣∣ ≤ LfLg Γ2(1 + σ) Γ2(1 + ρ + η) I1/r(r, αf ) I1/s(s, αg)
2
(
αfr + 1
)1/r(
αgs + 1
)1/s Γ2(ρ) Γ2(1 + σ + η)
tαf+αg , (4.5)
where
I(u, v) :=
(σ
ρ
)u{
B(uv + 2, u(ρ− 1) + 1) 2Ψ∗
1
[ (−u, 1), (uv + 2, η)(
(v + ρ− 1)u + 3, η
)
∣∣∣ 1− ρ
σ
]
+
1
(v + ρ− 1)u + 2 2Ψ∗
1
[ (−u, 1), (1, η)(
(v + ρ− 1)u + 3, η
)
∣∣∣ 1− ρ
σ
]}
;
(ii) for η ∈ (0, 1),
∣∣TS(f, g)
∣∣ ≤ LfLg Γ2(1 + σ) Γ2(1 + ρ + η)J 1/r(r, αf )J 1/s(s, αg)
2
(
αfr + 1
)1/r(
αgs + 1
)1/s Γ2(ρ) Γ2(1 + σ + η)
tαf+αg , (4.6)
where
J (u, v) :=
(σ
ρ
)ηu
B(uv + 2, u(ρ− 1) + 1) 2F1
[ −ηu, uv + 2
(v + ρ− 1)u + 3
∣∣∣ 1− ρ
σ
]
+
1
(v + ρ− 1)u + 2 2F1
[ −ηu, (v + ρ− 1)u + 2
(v + ρ− 1)u + 3
∣∣∣ 1− σ
ρ
]
.
P r o o f. Taking a = −η, b = ρ − σ, c = ρ, the conditions of Lemma are
ful�lled with 0 < σ < ρ, so by (4.1) we have
2F1
[ a, b
c
; x
]
<
{
σ/ρ + (1− σ/ρ)(1− x)η η > 1,
min
{
H, J1
}
η ∈ (0, 1),
(4.7)
where
H =
(
1− (1− σ/ρ)x
)η
,
J1 = (1− σ/ρ)(1− x)η+σ + (σ/ρ)(1− x)σ−ρ,
and for η + σ ≥ 1 it is min{H, J1} = H.
(i) η > 1. By (3.11) and (4.7), we conclude
∣∣TS(f, g)
∣∣ ≤ LfLgt
αf+αg Γ2(1 + σ) Γ2(1 + ρ + η)
2
(
αfr + 1
)1/r(
αgs + 1
)1/s Γ2(ρ) Γ2(1 + σ + η)
152 Journal of Mathematical Physics, Analysis, Geometry, 2012, vol. 8, No. 2
Andreev�Korkin Identity, Saigo Fractional Integration Operator
×
( 1∫
0
xαf r+1
{
(1− x)r(ρ−1)
(
σ/ρ + (1− σ/ρ)xη
)r
+ xr(ρ−1)
(
σ/ρ + (1− σ/ρ)(1− x)η
)r
}
dx
)1/r
×
( 1∫
0
xαgs+1
{
(1− x)s(ρ−1)
(
σ/ρ + (1− σ/ρ)xη
)s
+ xs(ρ−1)
(
σ/ρ + (1− σ/ρ)(1− x)η
)s
}
dx
)1/s
. (4.8)
Since
I1(r) :=
1∫
0
xα−1(1− x)β−1(1 + γxδ)rdx =
∞∑
n=0
(
r
n
)
γn
∞∫
0
xα+δn−1(1− x)β−1dx
= Γ(β)
∞∑
n=0
(−r)nΓ(α + δn)
Γ(α + β + δn)
(−γ)n
n!
= B(α, β)
∞∑
n=0
(−r)n(α)δn
(α + β)δn
(−γ)n
n!
= B(α, β) · 2Ψ∗
1
[ (−r, 1), (α, δ)
(α + β, δ)
∣∣∣ − γ
]
, (4.9)
where B(·, ·) denotes the Eulerian beta function, and because
I2(r) :=
1∫
0
xν−1(1 + γ(1− x)δ)rdx =
∞∑
n=0
(
r
n
)
γn
∞∫
0
xν−1(1− x)δndx
= Γ(ν)
∞∑
n=0
(−r)nΓ(1 + δn)
Γ(ν + 1 + δn)
(−γ)n
n!
=
1
ν
∞∑
n=0
(−r)n(1)δn
(ν + 1)δn
(−γ)n
n!
=
1
ν
· 2Ψ∗
1
[ (−r, 1), (1, δ)
(ν + 1, δ)
∣∣∣ − γ
]
,
for δ > 0, in both cases ∆ = 1+δ−1− δ = 0, therefore the series I1,2(r) converge
in the whole range of |γ| < ∇ = δδ · δ−δ = 1. Hence, the �rst integral in (4.8)
becomes
I(r, αf ) =
(σ
ρ
)r{
B(αfr + 2, r(ρ− 1) + 1) 2Ψ∗
1
[ (−r, 1), (αfr + 2, η)(
(αf + ρ− 1)r + 3, η
)
∣∣∣ 1− ρ
σ
]
+
1
(αf + ρ− 1)r + 2 2Ψ∗
1
[ (−r, 1), (1, η)(
(αf + ρ− 1)r + 3, η
)
∣∣∣ 1− ρ
σ
]}
;
Journal of Mathematical Physics, Analysis, Geometry, 2012, vol. 8, No. 2 153
D. Jankov and T.K. Pog�any
both series converge for |1− ρ/σ| < 1, that is, in the assumed range σ < ρ < 2σ.
By this conclusion the case (i) is proved.
(ii) η ∈ (0, 1), η + σ ≥ 1. In this case H and J1 possess the common tangent
x 7→ 1− η(1− σ/ρ)x at the origin. Being
H ′′(x) = η(η − 1)
(
1−σ
ρ
)2(
1− (1− σ/ρ)x
)η−2
< 0
J ′′1 (x) =
(
1−σ
ρ
)
(1− x)σ−ρ−2
[
(η + σ)(η + σ − 1)(1− x)η+ρ+σ(ρ + 1− σ)
]
> 0 ,
we clearly conclude that H is concave and J1 is convex in the unit interval. Thus,
it is min{H,J1} = H according to Carlson's Lemma too.
Without condition η + σ ≥ 1, as mutatis mutandis min{H,J1} ≤ H, we
conclude the case (ii) by (3.11).
Now in both cases η ∈ (0, 1), σ > 0 and by (4.7) we have
∣∣TS(f, g)
∣∣ ≤ LfLg Γ2(1 + σ) Γ2(1 + ρ + η)
2
(
αfr + 1
)1/r(
αgs + 1
)1/s Γ2(ρ) Γ2(1 + σ + η)
tαf+αg
×
( 1∫
0
xαf r+1
{
(1− x)r(ρ−1)
(
σ/ρ + (1− σ/ρ)x)ηr
+ xr(ρ−1)
(
1− (1− σ/ρ)x
)ηr
}
dx
)1/r
×
( 1∫
0
xαgs+1
{
(1− x)s(ρ−1)
(
σ/ρ + (1− σ/ρ)x)ηs
+ xs(ρ−1)
(
1− (1− σ/ρ)x
)ηs
}
dx
)1/s
, (4.10)
and the �rst integral in (4.10) is equal to
J (r, αf ) =
(σ
ρ
)ηr
B(αfr + 2, r(ρ− 1) + 1) 2F1
[ −ηr, αfr + 2
(αf + ρ− 1)r + 3
∣∣∣ 1− ρ
σ
]
+
1
(αf + ρ− 1)r + 2 2F1
[ −ηr, (αf + ρ− 1)r + 2
(αf + ρ− 1)r + 3
∣∣∣ 1− σ
ρ
]
, (4.11)
where both hypergeometric series converge in the range of 0 < σ < ρ < 2σ.
154 Journal of Mathematical Physics, Analysis, Geometry, 2012, vol. 8, No. 2
Andreev�Korkin Identity, Saigo Fractional Integration Operator
Indeed, we have
J (r, αf ) =
(σ
ρ
)ηr
I1(ηr) +
1∫
0
xα−1
(
1− (1− σ/ρ)x
)ηr
dx =
(σ
ρ
)ηr
I1(ηr) + I3(r) ,
when in (4.9) one speci�es α = αfr + 2, β = r(ρ− 1) + 1, γ = ρ/σ − 1 and δ = 1,
while further short calculation gives us
I3(r) =
∞∑
n=0
(
ηr
n
)
(−1)n(1− σ/ρ)n
(αf + ρ− 1)r + 2 + n
.
Using the transformation
1
A + n
=
Γ(A + n)
Γ(A + 1 + n)
=
1
A
· (A)n
(A + 1)n
,
where A = (αf + ρ− 1)r + 2, we conclude
I3(r) =
1
(αf + ρ− 1)r + 2
∞∑
n=0
(−ηr)n
(
(αf + ρ− 1)r + 2
)
n(
(αf + ρ− 1)r + 3
)
n
(1− σ/ρ)n
n!
=
1
(αf + ρ− 1)r + 2 2F1
[ −ηr, (αf + ρ− 1)r + 2
(αf + ρ− 1)r + 3
∣∣∣ 1− σ
ρ
]
.
So is the proof of (4.12).
Finally, let us present two more results in which we discuss the hypergeometric
kernel function appearing in the Saigo fractional integration operator: the �rst
with the integer value parameters ηr, ηs, where r, s form the conjugated H�older
exponents, the second estimating functions H, J1 in Carlson's Lemma by uniform
upper bound equal to 1 on the whole unit interval.
Corollary 4.1. Let r, s, r−1 + s−1 = 1, r > 1, be conjugated H�older pa-
rameters, η ∈ (0, 1), ηr, ηs ∈ N. Then for all f ∈ LipLf
(αf ), g ∈ LipLg
(αg),
0 < σ < ρ < 2σ we have
∣∣TS(f, g)
∣∣ ≤ LfLg Γ2(1 + σ) Γ2(1 + ρ + η)K1/r(r, αf )K1/s(s, αg)
2
(
αfr + 1
)1/r(
αgs + 1
)1/s Γ2(ρ) Γ2(1 + σ + η)
tαf+αg ,
(4.12)
where
K(u, v) :=
(σ
ρ
)ηu
B(uv + 2, u(ρ− 1) + 1)Pηu
(
1− ρ
σ
)
+
Qηu
(
1− σ/ρ
)
(v + ρ− 1)u + 2
,
Journal of Mathematical Physics, Analysis, Geometry, 2012, vol. 8, No. 2 155
D. Jankov and T.K. Pog�any
and
Pηu(z) = 2F1
[ −ηu, uv + 2
(v + ρ− 1)u + 3
∣∣∣ z
]
=
ηu∑
n=0
(−ηu)n(uv + 2)n(
(v + ρ− 1)u + 3
)
n
n!
zn,
Qηu(z) = 2F1
[ −ηu, (v + ρ− 1)u + 2
(v + ρ− 1)u + 3
∣∣∣ z
]
=
ηu∑
n=0
(−ηu)n
(
(v + ρ− 1)u + 2
)
n(
(v + ρ− 1)u + 3
)
n
n!
zn
are polynomials of degree ηu.
P r o o f. Because ηr, ηs ∈ N, the hypergeometric functions
2F1
[ −ηu, uv + 2
(v + ρ− 1)u + 3
∣∣∣ ·
]
,
2F1
[ −ηu, (v + ρ− 1)u + 2
(v + ρ− 1)u + 3
∣∣∣ ·
]
reduce to the polynomials Pηr(·), Qηs(·) of degrees ηr, ηs, respectively. The claim
now follows from Theorem 4.1, case (ii).
Corollary 4.2. Let r, s, r−1 + s−1 = 1, r > 1, be conjugated H�older parame-
ters. Then for all f ∈ LipLf
(αf ), g ∈ LipLg
(αg), 0 < σ < ρ < 2σ, η ∈ (0, 1),
1 + r(ρ− 1) > 0, 1 + s(ρ− 1) > 0 we have
∣∣TS(f, g)
∣∣ ≤ LfLg Γ2(1 + σ) Γ2(1 + ρ + η)N 1/r(r, αf )N 1/s(s, αg)
2
(
αfr + 1
)1/r(
αgs + 1
)1/s Γ2(ρ) Γ2(1 + σ + η)
tαf+αg ,
(4.13)
where
N (u, v) = B
(
2 + uv, 1 + u(ρ− 1)
)
+
1
u(v + ρ− 1) + 2
.
P r o o f. By virtue of the obvious estimate min{J1,H} ≤ 1, (4.13) is a direct
consequence of Theorem 3.1 and Lemma.
Acknowledgments. The authors are grateful to the anonymous reviewer for
the useful advices and suggestions which signi�cantly improved the quality of the
article, especially, remarking a lack in De�nition 3.1.
156 Journal of Mathematical Physics, Analysis, Geometry, 2012, vol. 8, No. 2
Andreev�Korkin Identity, Saigo Fractional Integration Operator
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