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CUBO A Mathematical Journal
Vol.17, No¯ 03, (01–14). October 2015

Right General Fractional Monotone Approximation

George A. Anastassiou

Department of Mathematical Sciences, University of Memphis,

Memphis, TN 38152, U.S.A.,

ganastss@memphis.edu

ABSTRACT

Here is introduced a right general fractional derivative Caputo style with respect to

a base absolutely continuous strictly increasing function g. We give various examples

of such right fractional derivatives for different g. Let f be p-times continuously dif-

ferentiable function on [a, b], and let L be a linear right general fractional differential

operator such that L (f) is non-negative over a critical closed subinterval J of [a, b].

We can find a sequence of polynomials Qn of degree less-equal n such that L (Qn) is

non-negative over J, furthermore f is approximated uniformly by Qn over [a, b] .

The degree of this constrained approximation is given by an inequality using the first

modulus of continuity of f(p). We finish we applications of the main right fractional

monotone approximation theorem for different g.

RESUMEN

Aqúı introducimos una derivada fraccional derecha general al estilo de Caputo con

respecto a una base de funciones absolutamente continuas estrictamente crecientes g.

Damos varios ejemplos de dichas derivadas fraccionales derechas para diferentes g.

Sea f una función p-veces continuamente diferenciable en [a, b], y sea L un operador

diferencial fraccional derecho general tal que L(f) es no-negativo en un subintervalo

cerrado cŕıtico J de [a, b]. Podemos encontrar una sucesión de polinomios L (Qn) de

grado menor o igual a n tal que L (Qn) es no-negativo en J, más aún f es aproximada

uniformemente por Qn en [a, b] . El grado de esta aproximación restringida es dada

por una desigualdad usando el primer módulo de continuidad de f(p). Concluimos con

aplicaciones del teorema principal de aproximación monótona fraccional derecha para

diferentes g.

Keywords and Phrases: Right Fractional Monotone Approximation, general right fractional

derivative, linear general right fractional differential operator, modulus of continuity.

2010 AMS Mathematics Subject Classification: 26A33, 41A10, 41A17, 41A25, 41A29.



2 George A. Anastassiou CUBO
17, 3 (2015)

1 Introduction and Preparation

The topic of monotone approximation started in [11] has become a major trend in approximation

theory. A typical problem in this subject is: given a positive integer k, approximate a given

function whose kth derivative is ≥ 0 by polynomials having this property.

In [4] the authors replaced the kth derivative with a linear ordinary differential operator of

order k.

Furthermore in [1], the author generalized the result of [4] for linear right fractional differential

operators.

To describe the motivating result here we need

Definition 1. ([5]) Let α > 0 and ⌈α⌉ = m, (⌈·⌉ ceiling of the number). Consider f ∈ Cm ([−1, 1]).

We define the right Caputo fractional derivative of f of order α as follows:

(

Dα1−f
)

(x) =
(−1)

m

Γ (m − α)

∫1

x

(t − x)
m−α−1

f(m) (t) dt, (1)

for any x ∈ [−1, 1], where Γ is the gamma function Γ (ν) =
∫
∞

0
e−ttν−1dt, ν > 0.

We set

D01−f (x) = f (x) , (2)

Dm1−f (x) = (−1)
m
f(m) (x) , ∀ x ∈ [−1, 1] . (3)

In [1] we proved

Theorem 1.1. Let h, k, p be integers, h is even, 0 ≤ h ≤ k ≤ p and let f be a real function,

f(p) continuous in [−1, 1] with modulus of continuity ω1
(

f(p), δ
)

, δ > 0, there. Let αj (x), j =

h, h + 1, ..., k be real functions, defined and bounded on [−1, 1] and assume for x ∈ [−1, 0] that

αh (x) is either ≥ some number α > 0 or ≤ some number β < 0. Let the real numbers α0 = 0 <

α1 < 1 < α2 < 2 < ... < αp < p. Here D
αj
1−

f stands for the right Caputo fractional derivative of f

of order αj anchored at 1. Consider the linear right fractional differential operator

L :=

k∑

j=h

αj (x)
[

D
αj
1−

]

(4)

and suppose, throughout [−1, 0] ,

L (f) ≥ 0. (5)

Then, for any n ∈ N, there exists a real polynomial Qn (x) of degree ≤ n such that

L (Qn) ≥ 0 throughout [−1, 0] , (6)

and

max
−1≤x≤1

|f (x) − Qn (x)| ≤ Cn
k−pω1

(

f(p),
1

n

)

, (7)

where C is independent of n or f.



CUBO
17, 3 (2015)

Right General Fractional Monotone Approximation 3

Notice above that the monotonicity property is only true on [−1, 0], see (5), (6). However the

approximation property (7) it is true over the whole interval [−1, 1] .

In this article we extend Theorem 1.1 to much more general linear right fractional differential

operators.

We use here the following right generalised fractional integral.

Definition 2. (see also [8, p. 99]) The right generalised fractional integral of a function f with

respect to given function g is defined as follows:

Let a, b ∈ R, a < b, α > 0. Here g ∈ AC ([a, b]) (absolutely continuous functions) and is

strictly increasing, f ∈ L
∞
([a, b]). We set

(

Iαb−;gf
)

(x) =
1

Γ (α)

∫b

x

(g (t) − g (x))
α−1

g′ (t) f (t) dt, x ≤ b, (8)

clearly
(

Iαb−;gf
)

(b) = 0.

When g is the identity function id, we get that Iαb−;id = I
α
b−, the ordinary right Riemann-

Liouville fractional integral, where

(

Iαb−f
)

(x) =
1

Γ (α)

∫b

x

(t − x)
α−1

f (t) dt, x ≤ b, (9)

(

Iαb−f
)

(b) = 0.

When g (x) = ln x on [a, b], 0 < a < b < ∞, we get

Definition 3. ([8, p. 110]) Let 0 < a < b < ∞, α > 0. The right Hadamard fractional integral of

order α is given by

(

Jαb−f
)

(x) =
1

Γ (α)

∫b

x

(

ln
y

x

)α−1 f (y)

y
dy, x ≤ b, (10)

where f ∈ L
∞

([a, b]) .

We mention

Definition 4. The right fractional exponential integral is defined as follows: Let a, b ∈ R, a < b,

α > 0, f ∈ L
∞

([a, b]). We set

(

Iαb−;exf
)

(x) =
1

Γ (α)

∫b

x

(

et − ex
)α−1

etf (t) dt, x ≤ b. (11)

Definition 5. Let a, b ∈ R, a < b, α > 0, f ∈ L
∞

([a, b]), A > 1. We introduce the right

fractional integral

(

Iαb−;Axf
)

(x) =
ln A

Γ (α)

∫b

x

(

At − Ax
)α−1

Atf (t) dt, x ≤ b. (12)



4 George A. Anastassiou CUBO
17, 3 (2015)

We also give

Definition 6. Let α, σ > 0, 0 ≤ a < b < ∞, f ∈ L
∞

([a, b]). We set

(

Kαb−;xσf
)

(x) =
1

Γ (α)

∫b

x

(tσ − xσ)
α−1

f (t) σtσ−1dt, x ≤ b. (13)

We introduce the following general right fractional derivative.

Definition 7. Let α > 0 and ⌈α⌉ = m, (⌈·⌉ ceiling of the number). Consider f ∈ ACm ([a, b])

(space of functions f with f(m−1) ∈ AC ([a, b])). We define the right general fractional derivative

of f of order α as follows

(

Dαb−;gf
)

(x) =
(−1)

m

Γ (m − α)

∫b

x

(g (t) − g (x))
m−α−1

g′ (t) f(m) (t) dt, (14)

for any x ∈ [a, b], where Γ is the gamma function.

We set

Dmb−;gf (x) = (−1)
m
f(m) (x) , (15)

D0b−;gf (x) = f (x) , ∀ x ∈ [a, b] . (16)

When g = id, then Dαb−f = D
α
b−;idf is the right Caputo fractional derivative.

So we have the specific general right fractional derivatives.

Definition 8.

Dαb−;ln xf (x) =
(−1)

m

Γ (m − α)

∫b

x

(

ln
y

x

)m−α−1 f(m) (y)

y
dy, 0 < a ≤ x ≤ b, (17)

Dαb−;exf (x) =
(−1)

m

Γ (m − α)

∫b

x

(

et − ex
)m−α−1

etf(m) (t) dt, a ≤ x ≤ b, (18)

and

Dαb−;Axf (x) =
(−1)

m
ln A

Γ (m − α)

∫b

x

(

At − Ax
)m−α−1

Atf(m) (t) dt, a ≤ x ≤ b, (19)

(

Dαb−;xσf
)

(x) =
(−1)

m

Γ (m − α)

∫b

x

(tσ − xσ)
m−α−1

σtσ−1f(m) (t) dt, 0 ≤ a ≤ x ≤ b. (20)

We mention

Theorem 1.2. (Trigub, [12], [13]) Let g ∈ Cp ([−1, 1]), p ∈ N. Then there exists real polynomial

qn (x) of degree ≤ n, x ∈ [−1, 1], such that

max
−1≤x≤1

∣

∣

∣
g(j) (x) − q(j)n (x)

∣

∣

∣
≤ Rpn

j−pω1

(

g(p),
1

n

)

, (21)

j = 0, 1, ..., p, where Rp is independent of n or g.



CUBO
17, 3 (2015)

Right General Fractional Monotone Approximation 5

In [2], based on Theorem 1.2 we proved the following useful here result

Theorem 1.3. Let f ∈ Cp ([a, b]), p ∈ N. Then there exist real polynomials Q∗n (x) of degree

≤ n ∈ N, x ∈ [a, b], such that

max
a≤x≤b

∣

∣

∣
f(j) (x) − Q∗(j)n (x)

∣

∣

∣
≤ Rp

(

b − a

2n

)p−j

ω1

(

f(p),
b − a

2n

)

, (22)

j = 0, 1, ..., p, where Rp is independent of n or g.

Remark 1.4. Here g ∈ AC ([a, b]) (absolutely continuous functions), g is increasing over [a, b],

α > 0.

Let g (a) = c, g (b) = d. We want to calculate

I =

∫b

a

(g (t) − g (a))
α−1

g′ (t) dt. (23)

Consider the function

f (y) = (y − g (a))
α−1

= (y − c)
α−1

, ∀ y ∈ [c, d] . (24)

We have that f (y) ≥ 0, it may be +∞ when y = c and 0 < α < 1, but f is measurable on [c, d].

By [9], Royden, p. 107, exercise 13 d, we get that

(f ◦ g) (t) g′ (t) = (g (t) − g (a))
α−1

g′ (t) (25)

is measurable on [a, b], and

I =

∫d

c

(y − c)
α−1

dy =
(d − c)

α

α
(26)

(notice that (y − c)
α−1

is Riemann integrable).

That is

I =
(g (b) − g (a))

α

α
. (27)

Similarly it holds

∫b

x

(g (t) − g (x))
α−1

g′ (t) dt =
(g (b) − g (x))

α

α
, ∀ x ∈ [a, b] . (28)

Finally we will use

Theorem 1.5. Let α > 0, N ∋ m = ⌈α⌉, and f ∈ Cm ([a, b]). Then
(

Dαb−;gf
)

(x) is continuous in

x ∈ [a, b], −∞ < a < b < ∞.

Proof. By [3], Apostol, p. 78, we get that g−1 exists and it is strictly increasing on [g (a) , g (b)].

Since g is continuous on [a, b], it implies that g−1 is continuous on [g (a) , g (b)]. Hence f(m) ◦ g−1

is a continuous function on [g (a) , g (b)] .



6 George A. Anastassiou CUBO
17, 3 (2015)

If α = m ∈ N, then the claim is trivial.

We treat the case of 0 < α < m.

It holds that

(

Dαb−;gf
)

(x) =
(−1)

m

Γ (m − α)

∫b

x

(g (t) − g (x))
m−α−1

g′ (t) f(m) (t) dt =

(−1)
m

Γ (m − α)

∫b

x

(g (t) − g (x))
m−α−1

g′ (t)
(

f(m) ◦ g−1
)

(g (t)) dt = (29)

(−1)
m

Γ (m − α)

∫g(b)

g(x)

(z − g (x))
m−α−1

(

f(m) ◦ g−1
)

(z) dz.

An explanation follows.

The function

G (z) = (z − g (x))
m−α−1

(

f(m) ◦ g−1
)

(z)

is integrable on [g (x) , g (b)], and by assumption g is absolutely continuous : [a, b] → [g (a) , g (b)].

Since g is monotone (strictly increasing here) the function

(g (t) − g (x))
m−α−1

g′ (t)
(

f(m) ◦ g−1
)

(g (t))

is integrable on [x, b] (see [7]). Furthermore it holds (see also [7]),

(−1)
m

Γ (m − α)

∫g(b)

g(x)

(z − g (x))
m−α−1

(

f(m) ◦ g−1
)

(z) dz =

(−1)
m

Γ (m − α)

∫b

x

(g (t) − g (x))
m−α−1

g′ (t)
(

f(m) ◦ g−1
)

(g (t)) dt (30)

=
(

Dαb−;gf
)

(x) , ∀ x ∈ [a, b] .

And we can write

(

Dαb−;gf
)

(x) =
(−1)

m

Γ (m − α)

∫g(b)

g(x)

(z − g (x))
m−α−1

(

f(m) ◦ g−1
)

(z) dz,

(

Dαb−;gf
)

(y) =
(−1)

m

Γ (m − α)

∫g(b)

g(y)

(z − g (y))
m−α−1

(

f(m) ◦ g−1
)

(z) dz. (31)

Here a ≤ y ≤ x ≤ b, and g (a) ≤ g (y) ≤ g (x) ≤ g (b), and 0 ≤ g (b) − g (x) ≤ g (b) − g (y) .

Let λ = z − g (x), then z = g (x) + λ. Thus

(

Dαb−;gf
)

(x) =
(−1)

m

Γ (m − α)

∫g(b)−g(x)

0

λm−α−1
(

f(m) ◦ g−1
)

(g (x) + λ) dλ. (32)

Clearly, see that g (x) ≤ z ≤ g (b), and 0 ≤ λ ≤ g (b) − g (x) .



CUBO
17, 3 (2015)

Right General Fractional Monotone Approximation 7

Similarly

(

Dαb−;gf
)

(y) =
(−1)

m

Γ (m − α)

∫g(b)−g(y)

0

λm−α−1
(

f(m) ◦ g−1
)

(g (y) + λ) dλ. (33)

Hence it holds
(

Dαb−;gf
)

(y) −
(

Dαb−;gf
)

(x) =
(−1)

m

Γ (m − α)
·

[∫g(b)−g(x)

0

λm−α−1
((

f(m) ◦ g−1
)

(g (y) + λ) −
(

f(m) ◦ g−1
)

(g (x) + λ)
)

dλ+

∫g(b)−g(y)

g(b)−g(x)

λm−α−1
(

f(m) ◦ g−1
)

(g (y) + λ) dλ

]

. (34)

Thus we obtain
∣

∣

(

Dαb−;gf
)

(y) −
(

Dαb−;gf
)

(x)
∣

∣ ≤
1

Γ (m − α)
·

[

(g (b) − g (x))
m−α

m − α
ω1

(

f(m) ◦ g−1, |g (y) − g (x)|
)

+ (35)

∥

∥f(m) ◦ g−1
∥

∥

∞,[g(a),g(b)]

m − α

(

(g (b) − g (y))
m−α

− (g (b) − g (x))
m−α

)

]

=: (ξ) .

As y → x, then g (y) → g (x) (since g ∈ AC ([a, b])). So that (ξ) → 0. As a result

(

Dαb−;gf
)

(y) →
(

Dαb−;gf
)

(x) , (36)

proving that
(

Dαb−;gf
)

(x) is continuous in x ∈ [a, b] .

2 Main Result

We present

Theorem 2.1. Here we assume that g (b) − g (a) > 1. Let h, k, p be integers, h is even, 0 ≤

h ≤ k ≤ p and let f ∈ Cp ([a, b]), a < b, with modulus of continuity ω1
(

f(p), δ
)

, 0 < δ ≤ b − a.

Let αj (x), j = h, h + 1, ..., k be real functions, defined and bounded on [a, b] and assume for

x ∈
[

a, g−1 (g (b) − 1)
]

that αh (x) is either ≥ some number α
∗ > 0, or ≤ some number β∗ < 0.

Let the real numbers α0 = 0 < α1 ≤ 1 < α2 ≤ 2 < ... < αp ≤ p. Consider the linear right general

fractional differential operator

L =

k∑

j=h

αj (x)
[

D
αj
b−;g

]

, (37)

and suppose, throughout
[

a, g−1 (g (b) − 1)
]

,

L (f) ≥ 0. (38)



8 George A. Anastassiou CUBO
17, 3 (2015)

Then, for any n ∈ N, there exists a real polynomial Qn (x) of degree ≤ n such that

L (Qn) ≥ 0 throughout
[

a, g−1 (g (b) − 1)
]

, (39)

and

max
x∈[a,b]

|f (x) − Qn (x)| ≤ Cn
k−pω1

(

f(p),
b − a

2n

)

, (40)

where C is independent of n or f.

Proof. of Theorem 2.1.

Here h, k, p ∈ Z+, 0 ≤ h ≤ k ≤ p. Let αj > 0, j = 1, ..., p, such that 0 < α1 ≤ 1 < α2 ≤ 2 <

α3 ≤ 3... < ... < αp ≤ p. That is ⌈αj⌉ = j, j = 1, ..., p.

Let Q∗n (x) be as in Theorem 1.3.

We have that

(

D
αj
b−;gf

)

(x) =
(−1)

j

Γ (j − αj)

∫b

x

(g (t) − g (x))
j−αj−1 g′ (t) f(j) (t) dt, (41)

and
(

D
αj
b−;g

Q∗n

)

(x) =
(−1)

j

Γ (j − αj)

∫b

x

(g (t) − g (x))
j−αj−1 g′ (t) Q∗n

(j) (t) dt, (42)

j = 1, ..., p.

Also it holds

(

D
j
b−;g

f
)

(x) = (−1)
j
f(j) (x) ,

(

D
j
b−;g

Q∗n

)

(x) = (−1)
j
Q∗(j)n (x) , j = 1, ..., p. (43)

By [10], we get that there exists g′ a.e., and g′ is measurable and non-negative.

We notice that
∣

∣

∣

(

D
αj
b−;g

f
)

(x) − D
αj
b−;g

Q∗n (x)
∣

∣

∣
=

1

Γ (j − αj)

∣

∣

∣

∣

∣

∫b

x

(g (x) − g (t))
j−αj−1 g′ (t)

(

f(j) (t) − Q∗(j)n (t)
)

dt

∣

∣

∣

∣

∣

≤

1

Γ (j − αj)

∫b

x

(g (x) − g (t))
j−αj−1 g′ (t)

∣

∣

∣
f(j) (t) − Q∗(j)n (t)

∣

∣

∣
dt

(22)
≤

1

Γ (j − αj)

(∫b

x

(g (x) − g (t))
j−αj−1 g′ (t) dt

)

Rp

(

b − a

2n

)p−j

ω1

(

f(p),
b − a

2n

)

(28)
=

(g (b) − g (x))
j−αj

Γ (j − αj + 1)
Rp

(

b − a

2n

)p−j

ω1

(

f(p),
b − a

2n

)

≤

(g (b) − g (a))
j−αj

Γ (j − αj + 1)
Rp

(

b − a

2n

)p−j

ω1

(

f(p),
b − a

2n

)

. (44)



CUBO
17, 3 (2015)

Right General Fractional Monotone Approximation 9

Hence ∀ x ∈ [a, b], it holds
∣

∣

∣

(

D
αj
b−;g

f
)

(x) − D
αj
b−;g

Q∗n (x)
∣

∣

∣
≤

(g (b) − g (a))
j−αj

Γ (j − αj + 1)
Rp

(

b − a

2n

)p−j

ω1

(

f(p),
b − a

2n

)

, (45)

and

max
x∈[a,b]

∣

∣

∣
D

αj
b−;g

f (x) − D
αj
b−;g

Q∗n (x)
∣

∣

∣
≤

(g (b) − g (a))
j−αj

Γ (j − αj + 1)
Rp

(

b − a

2n

)p−j

ω1

(

f(p),
b − a

2n

)

, (46)

j = 0, 1, ..., p.

Above we set D0b−;gf (x) = f (x), D
0
b−;gQ

∗
n (x) = Q

∗
n (x), ∀ x ∈ [a, b], and α0 = 0, i.e.

⌈α0⌉ = 0.

Put

sj = sup
a≤x≤b

∣

∣α−1h (x) αj (x)
∣

∣ , j = h, ..., k, (47)

and

ηn = Rpω1

(

f(p),
b − a

2n

)





k∑

j=h

sj
(g (b) − g (a))

j−αj

Γ (j − αj + 1)

(

b − a

2n

)p−j


 . (48)

I. Suppose, throughout
[

a, g−1 (g (b) − 1)
]

, αh (x) ≥ α
∗ > 0. Let Qn (x), x ∈ [a, b], be a real

polynomial of degree ≤ n, according to Theorem 1.3 and (46), so that

max
x∈[a,b]

∣

∣

∣
D

αj
b−;g

(

f (x) + ηn (h!)
−1

xh
)

−
(

D
αj
b−;g

Qn

)

(x)
∣

∣

∣
≤ (49)

(g (b) − g (a))
j−αj

Γ (j − αj + 1)
Rp

(

b − a

2n

)p−j

ω1

(

f(p),
b − a

2n

)

,

j = 0, 1, ..., p.

In particular (j = 0) holds

max
x∈[a,b]

∣

∣

∣

(

f (x) + ηn (h!)
−1

xh
)

− Qn (x)
∣

∣

∣
≤ Rp

(

b − a

2n

)p

ω1

(

f(p),
b − a

2n

)

, (50)

and

max
x∈[a,b]

|f (x) − Qn (x)| ≤ ηn (h!)
−1

(max (|a| , |b|))
h
+ Rp

(

b − a

2n

)p

ω1

(

f(p),
b − a

2n

)

= ηn (h!)
−1

max
(

|a|
h
, |b|

h
)

+ Rp

(

b − a

2n

)p

ω1

(

f(p),
b − a

2n

)

= (51)

Rpω1

(

f(p),
b − a

2n

)





k∑

j=h

sj
(g (b) − g (a))

j−αj

Γ (j − αj + 1)

(

b − a

2n

)p−j


(h!)
−1

max
(

|a|
h
, |b|

h
)



10 George A. Anastassiou CUBO
17, 3 (2015)

+Rp

(

b − a

2n

)p

ω1

(

f(p),
b − a

2n

)

≤

Rpω1

(

f(p),
b − a

2n

)

nk−p·









k∑

j=h

sj
(g (b) − g (a))

j−αj

Γ (j − αj + 1)

(

b − a

2

)p−j


(h!)
−1

max
(

|a|
h
, |b|

h
)

+

(

b − a

2

)p


 . (52)

We have found that

max
x∈[a,b]

|f (x) − Qn (x)| ≤ Rp

[(

b − a

2

)p

+ (h!)
−1

max
(

|a|
h
, |b|

h
)

·





k∑

j=h

sj
(g (b) − g (a))

j−αj

Γ (j − αj + 1)

(

b − a

2

)p−j






nk−pω1

(

f(p),
b − a

2n

)

, (53)

proving (40).

Notice for j = h + 1, ..., k, that

(

D
αj
b−;g

xh
)

=
(−1)

j

Γ (j − αj)

∫b

x

(g (t) − g (x))
j−αj−1 g′ (t)

(

th
)(j)

dt = 0. (54)

Here

L =

k∑

j=h

αj (x)
[

D
αj
b−;g

]

,

and suppose, throughout
[

a, g−1 (g (b) − 1)
]

, Lf ≥ 0. So over a ≤ x ≤ g−1 (g (b) − 1), we get

α−1h (x) L (Qn (x))
(54)
= α−1h (x) L (f (x)) +

ηn

h!

(

D
αh
b−;g

(

xh
)

)

+

k∑

j=h

α−1h (x) αj (x)
[

D
αj
b−;g

Qn (x) − D
αj
b−;g

f (x) −
ηn

h!
D

αj
b−;g

xh
] (49)

≥ (55)

ηn

h!

(

D
αh
b−;g

(

xh
)

)

−





k∑

j=h

sj
(g (b) − g (a))

j−αj

Γ (j − αj + 1)

(

b − a

2n

)p−j


Rpω1

(

f(p),
b − a

2n

)

(56)

(48)
=

ηn

h!

(

D
αh
b−;g

(

xh
)

)

− ηn = ηn

(

D
αh
b−;g

(

xh
)

h!
− 1

)

= (57)

ηn

(

1

Γ (h − αh) h!

∫b

x

(g (t) − g (x))
h−αh−1 g′ (t)

(

th
)(h)

dt − 1

)

=

ηn

(

h!

h!Γ (h − αh)

∫b

x

(g (t) − g (x))
h−αh−1 g′ (t) dt − 1

)

(28)
=



CUBO
17, 3 (2015)

Right General Fractional Monotone Approximation 11

ηn

(

(g (b) − g (x))
h−αh

Γ (h − αh + 1)
− 1

)

= (58)

ηn

(

(g (b) − g (x))
h−αh − Γ (h − αh + 1)

Γ (h − αh + 1)

)

≥

ηn

(

1 − Γ (h − αh + 1)

Γ (h − αh + 1)

)

≥ 0. (59)

Clearly here g (b) − g (x) ≥ 1.

Hence

L (Qn (x)) ≥ 0, for x ∈
[

a, g−1 (g (b) − 1)
]

. (60)

A further explanation follows: We know Γ (1) = 1, Γ (2) = 1, and Γ is convex and positive on

(0, ∞). Here 0 ≤ h − αh < 1 and 1 ≤ h − αh + 1 < 2. Thus

Γ (h − αh + 1) ≤ 1 and 1 − Γ (h − αh + 1) ≥ 0. (61)

II. Suppose, throughout
[

a, g−1 (g (b) − 1)
]

, αh (x) ≤ β
∗ < 0.

Let Qn (x), x ∈ [a, b] be a real polynomial of degree ≤ n, according to Theorem 1.3 and (46),

so that

max
x∈[a,b]

∣

∣

∣
D

αj
b−;g

(

f (x) − ηn (h!)
−1

xh
)

−
(

D
αj
b−;g

Qn

)

(x)
∣

∣

∣
≤ (62)

(g (b) − g (a))
j−αj

Γ (j − αj + 1)
Rp

(

b − a

2n

)p−j

ω1

(

f(p),
b − a

2n

)

,

j = 0, 1, ..., p.

In particular (j = 0) holds

max
x∈[a,b]

∣

∣

∣

(

f (x) − ηn (h!)
−1

xh
)

− Qn (x)
∣

∣

∣
≤ Rp

(

b − a

2n

)p

ω1

(

f(p),
b − a

2n

)

, (63)

and

max
x∈[a,b]

|f (x) − Qn (x)| ≤ ηn (h!)
−1

(max (|a| , |b|))
h
+ Rp

(

b − a

2n

)p

ω1

(

f(p),
b − a

2n

)

= ηn (h!)
−1

max
(

|a|
h
, |b|

h
)

+ Rp

(

b − a

2n

)p

ω1

(

f(p),
b − a

2n

)

, (64)

etc.

We find again that

max
x∈[a,b]

|f (x) − Qn (x)| ≤ Rp

[(

b − a

2

)p

+ (h!)
−1

max
(

|a|
h
, |b|

h
)

·



12 George A. Anastassiou CUBO
17, 3 (2015)





k∑

j=h

sj
(g (b) − g (a))

j−αj

Γ (j − αj + 1)

(

b − a

2

)p−j






nk−pω1

(

f(p),
b − a

2n

)

, (65)

reproving (40).

Here again

L =

k∑

j=h

αj (x)
[

D
αj
b−;g

]

,

and suppose, throughout
[

a, g−1 (g (b) − 1)
]

, Lf ≥ 0. So over a ≤ x ≤ g−1 (g (b) − 1), we get

α−1h (x) L (Qn (x))
(54)
= α−1h (x) L (f (x)) −

ηn

h!

(

D
αh
b−;g

(

xh
)

)

+

k∑

j=h

α−1h (x) αj (x)
[

D
αj
b−;g

Qn (x) − D
αj
b−;g

f (x) +
ηn

h!
D

αj
b−;g

xh
] (62)

≤ (66)

−
ηn

h!

(

D
αh
b−;g

(

xh
)

)

+





k∑

j=h

sj
(g (b) − g (a))

j−αj

Γ (j − αj + 1)

(

b − a

2n

)p−j


Rpω1

(

f(p),
b − a

2n

)

(67)

(48)
= −

ηn

h!

(

D
αh
b−;g

(

xh
)

)

+ ηn = ηn

(

1 −
D

αh
b−;g

(

xh
)

h!

)

= (68)

ηn

(

1 −
1

Γ (h − αh) h!

∫b

x

(g (t) − g (x))
h−αh−1 g′ (t)

(

th
)(h)

dt

)

=

ηn

(

1 −
h!

h!Γ (h − αh)

∫b

x

(g (t) − g (x))
h−αh−1 g′ (t) dt

)

(28)
=

ηn

(

1 −
(g (b) − g (x))

h−αh

Γ (h − αh + 1)

)

= (69)

ηn

(

Γ (h − αh + 1) − (g (b) − g (x))
h−αh

Γ (h − αh + 1)

)

(61)
≤

ηn

(

1 − (g (b) − g (x))
h−αh

Γ (h − αh + 1)

)

≤ 0. (70)

Hence again

L (Qn (x)) ≥ 0, ∀ x ∈
[

a, g−1 (g (b) − 1)
]

.

The case of αh = h is trivially concluded from the above. The proof of the theorem is now over.

We make



CUBO
17, 3 (2015)

Right General Fractional Monotone Approximation 13

Remark 2.2. By Theorem 1.5 we have that D
αj
b−;g

f are continuous functions, j = 0, 1, ..., p. Sup-

pose that αh (x) , ..., αk (x) are continuous functions on [a, b], and L (f) ≥ 0 on
[

a, g−1 (g (b) − 1)
]

is replaced by L (f) > 0 on
[

a, g−1 (g (b) − 1)
]

. Disregard the assumption made in the main theorem

on αh (x). For n ∈ N, let Qn (x) be the Q
∗
n (x) of Theorem 1.3, and f as in Theorem 1.3 (same as

in Theorem 2.1). Then Qn (x) converges to f (x) at the Jackson rate
1

np+1
([6], p. 18, Theorem

VIII) and at the same time, since L (Qn) converges uniformly to L (f) on [a, b], L (Qn) > 0 on
[

a, g−1 (g (b) − 1)
]

for all n sufficiently large.

3 Applications (to Theorem 2.1)

1) When g (x) = ln x on [a, b], 0 < a < b < ∞.

Here we would assume that b > ae, αh (x) restriction true on
[

a, b
e

]

, and

Lf =

k∑

j=h

αj (x)
[

D
αj
b−;ln x

f
]

≥ 0, (72)

throughout
[

a, b
e

]

.

Then L (Qn) ≥ 0 on
[

a, b
e

]

.

2) When g (x) = ex on [a, b], a < b < ∞.

Here we assume that b > ln (1 + ea), αh (x) restriction true on
[

a, ln
(

eb − 1
)]

, and

Lf =

k∑

j=h

αj (x)
[

D
αj
b−;ex

f
]

≥ 0, (73)

throughout
[

a, ln
(

eb − 1
)]

.

Then L (Qn) ≥ 0 on
[

a, ln
(

eb − 1
)]

.

3) When, A > 1, g (x) = Ax on [a, b], a < b < ∞.

Here we assume that b > logA (1 + A
a), αh (x) restriction true on

[

a, logA
(

Ab − 1
)]

, and

Lf =

k∑

j=h

αj (x)
[

D
αj
b−;Ax

f
]

≥ 0, (74)

throughout
[

a, logA
(

Ab − 1
)]

.

Then L (Qn) ≥ 0 on
[

a, logA
(

Ab − 1
)]

.

4) When σ > 0, g (x) = xσ, 0 ≤ a < b < ∞.

Here we assume that b > (1 + aσ)
1
σ , αh (x) restriction true on

[

a, (bσ − 1)
1
σ

]

, and

Lf =

k∑

j=h

αj (x)
[

D
αj
b−;xσ

f
]

≥ 0 (75)



14 George A. Anastassiou CUBO
17, 3 (2015)

throughout
[

a, (bσ − 1)
1
σ

]

.

Then L (Qn) ≥ 0 on
[

a, (bσ − 1)
1
σ

]

.

Received: April 2015. Accepted: July 2015.

References

[1] G.A. Anastassiou, Right Fractional Monotone Approximation, J. Applied Functional Analysis,

Vol. 10, No.’s 1-2 (2015), 117-124.

[2] G.A. Anastassiou, Left gereral fractional monotone approximation theory, submitted, 2015.

[3] T. Apostol, Mathematical Analysis, Addison-Wesley Publ. Co., Reading, Massachusetts, 1969.

[4] G.A. Anastassiou, O Shisha, Monotone approximation with linear differential operators, J.

Approx. Theory 44 (1985), 391-393.

[5] A.M.A. El-Sayed, M. Gaber, On the finite Caputo and finite Riesz derivatives, Electronic

Journal of Theoretical Physics, Vol. 3, No. 12 (2006), 81-95.

[6] D. Jackson, The Theory of Approximation, Amer. Math. Soc. Colloq., Vol. XI, New York,

1930.

[7] Rong-Qing Jia, Chapter 3. Absolutely Continous Functions, https://

www.ualberta.ca/˜rjia/Math418/Notes/Chap.3.pdf

[8] A.A. Kilbas, H.M. Srivastava and J.J. Trujillo, Theory and Applications of Fractional Differ-

ential Equations, vol. 204 of North-Holland Mathematics Studies, Elsevier, New York, NY,

USA, 2006.

[9] H.L. Royden, Real Analysis, second edition, Macmillan Publishing Co., Inc., New York, 1968.

[10] Anton R. Schep, Differentiation of Monotone Functions, people.

math.sc.edu/schep/diffmonotone.pdf.

[11] O. Shisha, Monotone approximation, Pacific J. Math. 15 (1965), 667-671.

[12] S.A. Teljakovskii, Two theorems on the approximation of functions by algebraic polynomials,

Mat. Sb. 70 (112) (1966), 252-265 [Russian]; Amer. Math. Soc. Trans. 77 (2) (1968), 163-178.

[13] R.M. Trigub, Approximation of functions by polynomials with integer coeficients, Izv. Akad.

Nauk SSSR Ser. Mat. 26 (1962), 261-280 [Russian].


	Introduction and Preparation
	Main Result
	Applications (to Theorem 2.1)