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Вестник КРАУНЦ. Физ.-мат. науки. 2020. Т. 33. № 4. C. 26-36. ISSN 2079-6641
MSC 26D15, 26D20, 34N05 Research Article
Assembling classical and dynamic inequalities accumulated on
calculus of time scales
Muhammad Jibril Shahab Sahir
Department of Mathematics, University of Sargodha, Sub-Campus Bhakkar, Bhakkar 30000, Pakistan
E-mail: jibrielshahab@gmail.com
In this paper, we present an extension of dynamic Renyi's inequality on time scales by using the time scale Riemann-Liouville type fractional integral. Furthermore, we find generalizations of the well-known Lyapunov's inequality and Radon's inequality on time scales by using the time scale Riemann-Liouville type fractional integrals. Our investigations unify and extend some continuous inequalities and their corresponding discrete analogues.
Keywords: time scales, fractional Riemann-Liouville integral, Renyi's inequality, Lyapunov's inequality, Radon's inequality.
DOI: 10.26117/2079-6641-2020-33-4-26-36
Original article submitted: 01.09.2020 Revision submitted: 10.10.2020
For citation. Sahir M. J. S. Assembling classical and dynamic inequalities accumulated on calculus of time scales. Vestnik KRAUNC. Fiz.-mat. nauki. 2020,33: 4,26-36. DOI: 10.26117/20796641-2020-33-4-26-36
The content is published under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0/deed.ru)
© Sahir M. J.S., 2020
1. Introduction
We introduce here some classical inequalities.
If xk > 0, yk > 0 for k = 1,2,...,n and X > 0 with X = 1, then
Ï-T (I,-Hi,»)"sÏ-TI,(1)
The inequality from (1) is called, in literature, Renyi's inequality as given in [10]. If xk > 0, yk > 0, k = 1,2,...,n and 0 < X1 < X2 < X3 < then
\ A3—Xi / \ A3— X2 / \ X2—Xi
n l\31 n n l\21 I XkyA2 W £ XkyA1 £ XkyA3) . (2)
vk=1 / \k=1 J \k=1 J
The inequality from (2) is called, in literature, Lyapunov's inequality as given in [12].
Funding. This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors
If > 0, yk > 0 for k = 1,2,...,n and X > 0, then
(3)
The inequality from (3) is called, in literature, Radon's inequality as given in [13].
We will unify and extend these results on time scales. The calculus of time scales was initiated by Stefan Hilger as given in [9]. A time scale is an arbitrary nonempty closed subset of the real numbers. The theory of time scales is applied to combine results in one comprehensive form. The three most popular examples of calculus on time scales are differential calculus, difference calculus, and quantum calculus, i.e., when T = R, T = N and T = qNo = {qt : t e No} where q > 1. The time scales calculus is studied as delta calculus, nabla calculus and diamond-a calculus. This hybrid theory is also widely applied on dynamic inequalities. The basic work on dynamic inequalities is done by Ravi Agarwal, George Anastassiou, Martin Bohner, Allan Peterson, Donal O'Regan, Samir Saker and many other authors.
In this paper, it is assumed that all considerable integrals exist and are finite and T is a time scale, a,b e T with a < b and an interval [a,b]T means the intersection of a real interval with the given time scale.
2. Preliminaries
We need here basic concepts of delta calculus. The results of delta calculus are adopted from monographs [6, 7].
For t e T, the forward jump operator a : T ^ T is defined by
The mapping ^ : T ^ R+ = [0, +<*>) such that ^(t) := a(t) -1 is called the forward graininess function. The backward jump operator p : T ^ T is defined by
The mapping v : T ^ R+ = [0, such that v(t) := t - p(t) is called the backward graininess function. If a(t) > t, we say that t is right-scattered, while if p(t) < t, we say that t is left-scattered. Also, if t < supT and a(t) = t, then t is called right-dense, and if t > infT and p(t) = t, then t is called left-dense. If T has a left-scattered maximum M, then Tk = T - {M}, otherwise Tk = T.
For a function f: T ^ R, the delta derivative fA is defined as follows: Let t e Tk. If there exists fA(t) e R such that for all e > 0, there is a neighborhood U of t, such that
for all s e U, then f is said to be delta differentiable at t, and /A(t) is called the delta derivative of f at t.
a(t) := inf{s e T : s > t}.
p(t) := sup{s e T : s < t}.
|f (a(t)) - f (s) - fA(t)(a(t) -s)| < e|a(t) -s|,
A function f: T ^ R is said to be right-dense continuous (rd-continuous), if it is continuous at each right-dense point and there exists a finite left-sided limit at every left-dense point. The set of all rd-continuous functions is denoted by Crd(T,R).
The next definition is given in [6, 7].
Definition 2.1. A function F : T ^ R is called a delta antiderivative of f : T ^ R, provided that FA(t) = f (t) holds for all t e Tk. Then the delta integral of f is defined by
i b
/ f (t)At = F(b) -F(a).
J a
The following results of nabla calculus are taken from [5, 6, 7].
If T has a right-scattered minimum m, then Tk = T — {m}, otherwise Tk = T. A function f: Tk ^ R is called nabla differentiable at t e Tk, with nabla derivative fv(t), if there exists fV(t) e R such that given any e > 0, there is a neighborhood V of t, such that
|f (p(t)) — f (s) — f V(t)(p(t) — s) | < e|p(t) — s|,
for all s e V.
A function f: T ^ R is said to be left-dense continuous (ld-continuous), provided it is continuous at all left-dense points in T and its right-sided limits exist (finite) at all right-dense points in T. The set of all ld-continuous functions is denoted by C1d(T,R).
The next definition is given in [5, 6, 7].
Definition 2.2. A function G : T ^ R is called a nabla antiderivative of g : T ^ R, provided that Gv(t) = g(t) holds for all t e Tk. Then the nabla integral of g is defined by
i b
/ g(t)Vt = G(b) — G(a).
a
We need the following results.
Theorem 2.1 ([1]). Let w,f,g e Crd ([a,b]T,R). If p + 1 = 1 with p > 1, then
P q
rb frb \ \ frb \ q
/ |w(x)||f(x)g(x)|Ax W/ |w(x)||f(x)|PAx / |w(x)||g(x)|qA^ . (4)
Ja \Ja / \Ja /
If 1p + q = 1 with p < 0 or q < 0, then inequality (4) is reversed.
Inequality (4) is called dynamic Rogers-Holder's inequality on time scales. We consider the Kantorovich's ratio defined by
(h +1)2
K(h) := ,, ; , where h > 0. w 4h
The function K is decreasing on (0,1) and increasing on [1, +<^), K(h) > 1 for any h > 0 and K(h) = K (1) for any h > 0.
The following reverse of Young's inequality [11] in terms of Kantorovich's ratio holds
a b s /a\ i i
< K^ apbq, (5)
p q b
where a, b > 0, p + q = 1 with p > 1 and 8 = ma^ p, q
p q I p q
The following definition is taken from [2, 4].
Definition 2.3. For a > 1, the time scale A-Riemann-Liouville type fractional integral for a function f e Crd is defined by
Iff (t )= fha-1 (t, a (T))f (t )At , (6)
J a
which is an integral on [a,t)T, as given in [8] and ha : T x T ^ R, a > 0 are the coordinate wise rd-continuous functions, such that h0(t,s) = 1,
ha+1 (t, s) = fha (t , s)At , Vs, t e T. (7)
Js
Notice that
la1 f (t) = ff (T )At ,
■J a
which is absolutely continuous in t e [a,b]T, see [8]. The following definition is taken from [3, 4].
Definition 2.4. For a > 1, the time scale V-Riemann-Liouville type fractional integral for a function f e C1d is defined by
jaa f (t)= A a -1 (t, p (t)) f (t )Vt , (8)
a
which is an integral on (a, t]T, as given in [8] and ha : T x T ^ R, a > 0 are the coordinate wise ld-continuous functions, such that h0(t,s) = 1,
ha+1 (t,s) = jfha(t,s)VT, Vs,t e T. (9)
Notice that
Ja f (t) = ff (T )VT,
a
which is absolutely continuous in t e [a,b]T, see [8].
3. Renyi's inequality
In order to present our main results, first we give a simple proof for an extension of dynamic Renyi's inequality on time scales by using the time scale A-Riemann-Liouville type fractional integral.
Theorem 3.1. Let w, f,g G Crd ([a,b]T,R) be A-integrable functions and ha_i(.,.) > 0. If X g R+ _ {0,1}, then for a > 1
1 (Ia (|w(x)||f(x)|))X (Ia (|w(x)||g(x)|))1-X
X - 1 a v^a
ra I 11 \ \ 11_ X
- X1!Ia (|wCx)||f (x)|X |g(x)|1-^ . (10)
X -1
Proof. Case (1). If X > 1, then dynamic Rogers-Holder's inequality (4) becomes
1 X-1
|w(y)|| f (y)g(y) |Ay < ( I |w(y)|| f(y)|X Ay) V f |w(y)||g(y)|I- Ay) X . (11)
ia \Ja / \Ja /
X—1
Replacing |g(y)| by |g(y)|^^ in inequality (11), we obtain
1 X—1
f |w(y)||/(y)||g(y)|1 Ay < ( f |w(y)||/(y)|XAy) V f |w(y)||g(y)|Ay) X . (12)
J a \J a / \Ja /
Taking power X > 1 on both sides of inequality (12) and replacing |w(y)| by ha—1(x, a(y)) |w(y)| with ha—1(x, a(y)) > 0, Vx e [a,b]T, we get
X1
ha_i(x,a(y))|w(y)||/(y)||g(y)|^Ay) ( £ ha_i(x,a(y))|w(y)||g(y)|Ay
r x
rx
1- X
< /xha-i(x, a(y))|w(y)||/(y)|XAy. (13)
a
1X
Replacing |/(y)| by |/(y)||g(y)|^ in inequality (13) and multiplying both sides by x1i, we get the desired claim.
Case (2). If X e (0,1), then by using dynamic Rogers-Holder's inequality (4) for p = 1 > 1 and q = ilx > 1, we obtain
X / /»x \ 1—X
/ |w(y)||/(y)|X|g(y)|1—XAy < / |w(y)||/(y)|Ay ^ i I |w(y)||g(y)|Ay) . (14)
J a \Ja / \Ja /
Replacing |w(y)| by ha—1(x, a(y))|w(y)| with ha—1(x, a(y)) > 0, Vx e [a,b]T and multiplying both sides of inequality (14) by x1i, we get the desired claim. Thus, the proof of Theorem 3.1 is now complete. □
Next, we give an extension of dynamic Renyi's inequality on time scales by using the time scale V-Riemann-Liouville type fractional integral.
Theorem 3.2. Let w,/,g e C1d ([a,b]T,R) be V-integrable functions and ha—1(.,.) > 0. If X e R+ — {0,1}, then for a > 1
1 „a ,, , N , , „ , n 'n \ X / ,za \ ..( -i \ \ 1 X
(Jaa (|w(x)||/(x) |)) ( Ja (|w(x)||g(x)|))
<
1
Jaa (|w(x)||/(x)r|g(x)|1—^ . (15)
Proof. Similar to the proof of Theorem 3.1. □
Remark 3.1. Let a = 1, T = Z, a = 1, x = b = n + 1, w = 1, /(k) = xk e [0, +<*>) and g(k) = yk e [0, for k = 1,2,...,n. Then inequality (10) reduces to (1).
X
x
4. Lyapunov's inequality
In this section, we give a simple proof for an extension of dynamic Lyapunov's inequality and its reverse version via time scales by using the time scale A-Riemann-Liouville type fractional integral.
Theorem 4.1. Let w,/,g,h e Crd ([a,b]T,R — {0}) be A-integrable functions and ha—1(.,.) > 0. Further assume that |/|Xl(X3—■X2)|g|X2(Xl—X3)|h|X3(X2—Xl) = M for X1,X2,X3 e R, where M is a positive real number.
(1) If A1 < A2 < A3, then for a > 1
Ia (V(x)||f(x)|Ai))A3-"2 (laa (|w(x)||g(x)|A2
X (laa (|w(x)||h(x)|A^)A2-Al > M. (16)
(2) If A2 < A1 < A3, then for a > 1
Iaa (|w(x)||f(X)|A1 (1aa (|w(x)||g(x)|A2^
X (laa (|w(x)||h(x)|A^)A2-A1 < M. (17)
Proof. Case (1). We note that X1 - X2 + X3 - X1 + X2 - X3 = 0. Set P = X|—X1 > 1,
Q = ^ > L Then P + Q = 1. 3 2
Applying Rogers-Holder's inequality (4), we get
|w(y)||f(y)h(y)|Ay < ( / |w(y)||f(y)|PA^W / |w(y)||h(y)|QAy . (18)
1 i P / fx ^ \ Q
Replacing | f (y)| by | f (y)| p and |h(y)| by |h(y)| Q in inequality (18), we obtain
^2 MA3^
|w(y) 11 f (y) | V |h(y) | ^ A3—A1 7 Ay
A3— A2 A2 —A1 •»x \ a _a f cx \ A _a
< ( I |w(y)||f(y)|A1 Ay 3— 1 / |w(y)||h(y)|A3Ay ) 3— 1 . (19)
a
Replacing |w(y)| by ha-1(x,a(y))|w(y)| with ha-1(x,a(y)) > 0, Vx e [a,b]T and taking power (X3 -X1) > 0 on both sides of inequality (19), we get
fx X (X3-X2\ X (X2-X1A \ X3-X1
' ha-1 (x, a(y)) |w(y) 11f(y) |MX3-XJ |h(y) |XHX2-X1 /Ay
a
/ rx \ A3-A2
< H ha-i(x, a(y))|w(y)||f(y)|AlAy)
/ rx \ A2-A1
xH ha-i(x, a(y))|w(y)||h(y)|A3Ayj . (20)
Using the condition that
|f |A1 (A3-A2) |g|A2(A1-A3) |h|X3(X2-Xl) _ M
for A1 < A2 < A3, where M is a positive real number, the inequality (20) takes the form
fX 1 \ A3-A1
la ha-1(x, a(y))|w(y) |M|g(y)|A2Ay
/ rx , \ A3—A2
ha—1(x, a(y))|w(y)||f(y)|A1 AyJ
/ rx \ A2—A1
xH ha—1 (x, a(y))|w(y)||h(y)|A3Ayj . (21)
x
This directly yields (16).
Similarly, we can prove the Case (2) by applying reverse Rogers-Holder's inequality. Thus, the proof of Theorem 4.1 is now complete. □
Next, we present an extension of dynamic Lyapunov's inequality and its reverse version via time scales by using the time scale V-Riemann-Liouville type fractional integral.
Theorem 4.2. Let w,f,g,h G C1d([a,b]T,R — {0}) be V-integrable functions and ha-i(.,.) > 0. Further assume that |fjAi(A3—A2)|g|'2('1— A3)|h|A3(A2—Al) = M for A2,A3 g R, where M is a positive real number.
(1) If A1 < A2 < A3, then for a > 1
( Ja" (|W(X)||f(x)|A ))A3-"2 ( Jfla (|w(x)||g(x)|'2j)'1-'3
X (Jaa (|w(x)||h(x)|A3))A2-A1 > M. (22)
(2) If A2 < A1 < A3, then for a > 1
( Ja" (|w(x)||f(x)|A ))A3-'2 ( Ja" (|w(x)||g(x)|'2))'1-A3
x (ja (|w(x)||h(x)|'3))A2-A1 <M. (23)
Proof. Similar to the proof of Theorem 4.1. □
Remark 4.1. Let a = 1, T = Z, a = 1, x = b = n + 1, w(k) = xk G (0, and f (k) = g(k) = h(k) = yk G (0, for k = 1,2,...,n. Then inequality (16) reduces to (2).
5. Radon's inequality
In order to conclude our main results, now we present an extension of dynamic Radon's inequality by using Kantorovich's ratio and the time scale A-Riemann-Liouville type fractional integral. Extensions of dynamic Radon's inequality are also proved in [14, 15, 16].
Theorem 5.1. Let w, f,g G Crd ([a,b]T,R — {0}) be A-integrable functions and ha-i(.,.) > 0. If X > 0 and y> 1, then for a > 1
I |w(x)||f(x)|X+A (Ia (k * (tfp )l»M||f (x)llg(x-)|Y -1 ))X
|g(X)|X - (1° (|w(x)||g(x)|Y))X+Y—1 , ( '
where
A(x) = Ia (^f^), "(x) = Ia (|w(x) | |g(x) |Y), Vx G [a, b]T, 8 = max j^, ^}
and K(.) is the Kantorovich's ratio.
~ I+Y
Proof. Let p = X + y > 1 and q = X++y 1 > 1. Setting
*0) = fx|W(/)!I/(/)|P and *(>) = |w(y)||g(y)|q
fa |w(y)||/(y)|pAy ^ ¡X |w(y)||g(y)|qAy
32
on [a,x]T, Vx e [a,b]T. From Young's inequality (5), we get
|w(y)||f (y)|p , |w(y)||g(y)|q ^ K' (|w(y)^f (y)g(y)|
P (fax |w(y)|| f(y)| pAy)+ q (fa |w(y)||g(y)|qAy) < (^(x)) j (i^)) 1 , (25)
where A1 (x) = fx |w(y) 11f (y) |PAy and l (x) = fx |w(y) | |g(y) |qAy. By integrating both sides of inequality (25) over y from a to x, we obtain
J < fa K5 (fgff )|w(y)||f(y)g(y)|Ay (26)
(fx |w(y) 11f (y) |jAy)1 (fx |w(y) | |g(y) |qAy)q.
Thus
¡■x \1 / rx \ q
/ |w(y)||/(y)|pAyJ lj |w(y)||g(y)|qAy
£ iv (Ii® |w(y)||f(y)g(y)|Ay. (27)
Replacing |f (y)| and |g(y)| by |F(y)| and |G(y)| in (27), respectively, we have
. \ / x X \ X+Y-1
|w(y)||F(y)|X+YAy) X+r[ j* |w(y)||G(y)|^ Ay)
< fxKs ( l2(x)|F(y)|XX+Y Vw(y)||F(y)G(y)|Ay, (28) Ja \A2(x)|G(y)| )
where A2(x) = fx |w(y)||F(y)|X+YAy and ^(x) = fx |w(y)||G(y)|Ay.
fi \ Xi" X+Y-1 1
Replacing |F(y)| by gg X+Y and |G(y)| by |f(y)|"W|g(y)|^ in (28), we obtain
1 X+Y-1
|w(y)(y)(y)| Ay) X+Y( f |w(y)||f (y)||g(y)|X+Y-1 A^ X+Y
'a |X WH / \Ja
< /xK5 (-l3(x) X+Y I |w(y)||f(y)|Ay, (29)
Ja \A3(x)|g(y)| X++-1
where A3 (x) = fax lwiy)f)iAy and I3M = fx |w(y)|| f (y)||g(y)| ^ Ay. Taking power X + 7 on both sides of inequality (29), we get
X+y
, _ v/axK5( -^^ ) |w(y)||f(y)|Ay '
^MMM \^ \ \A3(x)|g(y)|X+W_/ (30)
ig(y) Ay < ( 1 u+Y- . ()
|g(y)| fx |w(y)||f (y)||g(y)| X+Y-1 Ay)
Replacing |g(y)| by h«—i (x, a (y)) > 0, Vx g
A+y—i _ 1
and |w(y)| by ha—i(x, a(y))|w(y)||g(y)|Y 1 with
¿M
f (y)
a,in (30), we get
f xha-1(x, a (y))|w(y)|| f (y)|A+Y Ay
j a |g(y)|A y
(/qxha-1 (x, a(y))K8 (;;jx)|f((yy)p) |w(y)11f (y)| |g(y) |Y—1 Ay)'+' < t-! i , (31)
(/axha-1 (x, a (y)) | w (y) || g (y) | YAy)A+Y -1
where A(x) = 1^^W^^Ay, fl(x) = /flxha-1(x,a(y))|w(y)||g(y)|YAy. Thus (24) follows from (31). □
Theorem 5.2. Let w,f,g G C1d ([a,b]T,R - {0}) be V-integrable functions and ha-1(.,.) > 0. If A > 0 and y> 1, then for a > 1
•a ^ (Ï |w(x)|| f(x)|Ig(x)IY-
/a( |w(x)||f (x)|A+A lAC^J|w(x)||f (x)||g(x)|r-^ (32)
|g(x)|A 7" (ja (|w(x)||g(x)|Y ))A+Y -1 , ( )
where A(x) = Jaa ('^gg/+'), fl(x) = Jaa (|w(x)||g(x)|Y), Vx G [a, b]x, 8 = max j , A++-11, and K(.) is the Kantorovich's ratio. Proof. Similar to the proof of Theorem 5.1. □
Remark 5.1. Let a = 1, T = Z, a = 1, x = b = n + 1, w = 1, f (k) = xk G (0, +~) and g(k) = yk g (0, +~) for k = 1,2,...,n. Then discrete version of inequality (24) reduces to
s iMifKT, (33)
n /+Y n Y
where A = £ and ^ = £ yk.
k=1 yk k=1
The inequality (33) is obtained by using Kantorovich's ratio and is a reverse version of inequality (3) for y = l.
Competing interests. The author declares that there are no conflicts of interest regarding authorship and publication.
Contribution and Responsibility. The author contributed to this article. The author is solely responsible for providing the final version of the article in print. The final version of the manuscript was approved by the author.
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Вестник КРАУНЦ. Физ.-Мат. Науки. 2020. Т. 33. №. 4. С. 26-36. ISSN 2079-6641
УДК 519.644 Научная статья
Объединение классических и динамических неравенств, возникающих при анализе временных масштабов
М.Дж.Ш. Сахир
Университет Саргодха, кафедра математика, суб-кампус Бхаккар, Бхаккар, 30000, Пакистан
E-mail: jibrielshahab@gmail.com
В этой статье мы представляем расширение динамического неравенства Реньи на шкалы времени с помощью дробного интеграла типа Римана-Лиувилля. Кроме того, мы находим обобщения хорошо известного неравенства Ляпунова и неравенства Радона на шкалах времени с помощью дробных интегралов типа Римана-Лиувилля на шкале. Наши исследования объединяют и расширяют некоторые непрерывные неравенства и соответствующие им дискретные аналоги.
Ключевые слова: шкалы времени, дробный интеграл Римана-Лиувилля, неравенство Реньи, неравенство Ляпунова, неравенство Радона.
DOI: 10.26117/2079-6641-2020-33-4-26-36
Поступила в редакцию: 01.09.2020 В окончательном варианте: 10.10.2020
Для цитирования. Сахир М. Дж. Ш. Объединение классических и динамических неравенств, возникающих при анализе временных масштабов // Вестник КРАУНЦ. Физ.-мат. науки. 2020. Т. 33. № 4. C. 26-36. DOI: 10.26117/2079-6641-2020-33-4-26-36
Конкурирующие интересы. Конфликтов интересов в отношении авторства и публикации нет.
Авторский вклад и ответсвенность. Автор участвовал в написании статьи и полностью несет ответственность за предоставление окончательной версии статьи в печать.
Контент публикуется на условиях лицензии Creative Commons Attribution 4.0 International (https://creativecommons.Org/licenses/by/4.0/deed.ru)
© Сахир М. Д.Ш., 2020
Финансирование. Исследование выполнялось без финансирования
