Faculty of Sciences and Mathematics University of Niˇs
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Filomat 20:1 (2006), 23–32
SOME INEQUALITIES FOR FUNCTIONS OF BOUNDED VARIATION WITH APPLICATIONS TO LANDAU TYPE RESULTS S.S. DRAGOMIR Abstract. Some inequalities for functions of bounded variation that provide reverses for the inequality between the integral mean and the p−norm for p ∈ [1, ∞] are established. Applications related to the celebrated Landau inequality between the norms of the derivatives of a function are also pointed out.
1. Introduction The following inequality holding on finite intervals is well known. If f : [a, b] → R is essentially bounded, then f is integrable on [a, b] and ¯Z ¯ ¯ 1 ¯¯ b f (t) dt¯¯ ≤ kf k[a,b],∞ (1.1) ¯ b−a a where kf k[a,b],∞ := ess supt∈[a,b] |f (t)| . The corresponding version in terms of p−norms, is the following H¨older type inequality ¯Z b ¯ ¯ ¯ 1 ¯ p ≥ 1, f (t) dt¯¯ ≤ kf k[a,b],p , (1.2) 1 ¯ 1− (b − a) p a 2000 Mathematics Subject Classification. Primary 26D15; Secondary 26D10. Key words and phrases. Functions of bounded variation, Landau type inequalities, Inequalities for p−norms. 1 Received: February 13, 2006 23
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S.S. DRAGOMIR
provided f ∈ Lp [a, b] , where µZ kf k[a,b],p :=
b
p
¶ p1
|f (t)| dt
,
p ≥ 1.
a
In the first part of this paper we point out some reverse inequalities for (1.1) and (1.2) in the case of functions of bounded variation. These results are then employed in obtaining some Landau type inequalities. For the latter, recall that if I = R+ or I = R and if f : I → R is twice differentiable with f, f 00 ∈ Lp (I) , p ∈ [1, ∞] , then f 0 ∈ Lp (I) . Moreover, there exists a constant Cp (I) > 0 independent of the function f, such that 1 ° ° 0° °1 °f ° ≤ Cp (I) kf k 2 °f 00 ° 2 , (1.3) p,I p,I p,I where k·kp,I is the p−norm on the interval I. The investigation of such inequalities was initiated by E. Landau [8] in 1914. He considered the case p = ∞ and proved that √ (1.4) C∞ (R+ ) = 2 and C∞ (R) = 2, are the best constant for which (1.3) holds. For some classical and recent results related to Landau inequality, see [1],[4] and [5]-[11]. 2. Some Reverse Inequalities on Bounded Intervals The following result for functions of bounded variation holds. Theorem 1. Let f : [a, b] → R be a function of bounded variation on [a, b] . Then ¯Z ¯ _ ¯ b 1 ¯¯ b ¯+ f (t) dt (2.1) kf k[a,b],∞ ≤ (f ) . ¯ a b−a¯ a W The multiplicative constant 1 in front of ba (f ) cannot be replaced by a smaller quantity. Proof. We apply the following Ostrowski type inequality obtained by the author in [2] (see also [3]): ¯ ¯# ¯ ¯ " Z b ¯x − a+b ¯ _b ¯ ¯ 1 1 2 ¯f (x) − (2.2) f (t) dt¯¯ ≤ + (f ) ¯ a b−a a 2 b−a for any x ∈ [a, b] . The constant 12 is best possible in the sense that it cannot be replaced by a smaller quantity. Taking the supremum in (2.2) over x ∈ [a, b], we get " ¯ ¯# ° ° Z b ¯x − a+b ¯ _b ° ° 1 1 2 °f − (2.3) ≤ sup (f ) f (t) dt° + ° ° a b−a a b−a x∈[a,b] 2 [a,b],∞ _b (f ) . = a
INEQUALITIES FOR FUNCTIONS OF BOUNDED VARIATION
25
Now, by the triangle inequality applied for the sup-norm k·k∞ , we get ° ° ¯ ¯ Z b Z b ° ° ¯ 1 ¯ 1 ° ° ¯ kf k[a,b],∞ ≤ °f − f (t) dt° f (t) dt¯¯ +¯ b−a a b−a a [a,b],∞ ¯Z b ¯ _ ¯ b 1 ¯¯ ¯+ f (t) dt (f ) ≤ ¯ ¯ a b−a a and the inequality (2.1) is proved. To prove the sharpness of the constant 1, assume that the following inequality holds ¯Z ¯ _b ¯ 1 ¯¯ b ¯+C (2.4) kf k[a,b],∞ ≤ f (t) dt (f ) ¯ a b−a¯ a with a C > 0. Consider the function f0 : [a, b] → R, 0, t ∈ [a, b) f0 (t) = 1, t = b. Then f0 is of bounded variation on [a, b] and Z b kf0 k[a,b],∞ = 1, f0 (t) dt = 0 and a
_b a
(f0 ) = 1.
For this choice, (2.4) becomes C ≥ 1, proving the sharpness of the constant. ¤ The corresponding result for p−norms, where p ≥ 1, is embodied in the following theorem. Theorem 2. Let f : [a, b] → R be a function of bounded variation on [a, b] . Then for p ≥ 1 one has the inequality ¢1 1 ¡ ¯Z b ¯ ¯ ¯ 1 (b − a) p 2p+1 − 1 p _b 1 ¯ (2.5) kf k[a,b],p ≤ f (t) dt¯¯+ · (f ) . 1 1− 1 ¯ a 2 (b − a) p a (p + 1) p The constant 12 is best possible in the sense that it cannot be replaced by a smaller quantity. Proof. Taking the p−norm in (2.2), we deduce ° ° Z b _b ° ° °f − 1 ° ≤ (f ) Ip , f (t) dt ° ° a b−a a [a,b],p where
ÃZ " b Ip :=
a
¯ ¯ #p ! p1 ¯ 1 ¯x − a+b 2 + , dx 2 b−a
p ≥ 1.
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S.S. DRAGOMIR
We observe that ÃZ a+b " #p #p ! 1 Z b " p a+b 2 1 1 x − a+b 2 −x 2 Ip := + dx + + dx a+b 2 b−a 2 b−a a 2 "Z a+b # Z b 2 1 p = (b − x) dx + (x − a)p dx a+b b−a a 2 ¢1 1 ¡ p+1 (b − a) p 2 −1 p = , p ≥ 1. 1 2 (p + 1) p Using the triangle inequality for the p−norm k·kp , we get ° ° ° ° Z b Z b ° ° ° 1 ° 1 ° ° ° f (t) dt° +° f (t) dt° kf k[a,b],p ≤ °f − ° b−a a b−a a [a,b],p [a,b],p 1 ¢ 1 ¡ ¯ ¯ Z b ¯ 1 ¯ (b − a) p 2p+1 − 1 p _b 1 f (t) dt¯¯ ≤ (f ) + (b − a) p ¯¯ 1 a b−a a 2 (p + 1) p and the inequality (2.5) is obtained. Now, assume that (2.5) holds with a constant D > 0 instead of 12 , i.e., (2.6) ¢1 1 ¡ ¯Z b ¯ p 2p+1 − 1 p _b ¯ ¯ (b − a) 1 ¯ f (t) dt¯¯ + D · kf k[a,b],p ≤ (f ) . 1 1− 1 ¯ a (b − a) p a (p + 1) p Consider the function f0 : [a, b] → R with a = 0 and b > 1 given by 0, if t ∈ [0, b − 1] f0 (t) = 1, if t ∈ (b − 1, b]. This function is of bounded variation on [a, b] and Z b _b kf k[a,b],p = 1, f (t) dt = 1 and (f ) = 1 a
a
and then, by (2.6), we deduce 1≤
1
+D
1− p1
b
¢1 1 ¡ b p 2p+1 − 1 p 1
(p + 1) p
,
b > 1, p ≥ 1
giving (2.7)
1− p1
b
≤1+D·b
Denote q :=
¡ p+1 ¢1 2 −1 p 1
(p + 1) p
¡ p+1 ¢1 2 −1 p 1
(p + 1) p
.
.
INEQUALITIES FOR FUNCTIONS OF BOUNDED VARIATION
27
Then
¡ ¢ ln 2p+1 − 1 − ln (p + 1) ln q = . p We observe, by L’Hospital theorem that " ¡ ¢# £ ¡ p+1 ¢¤0 ln 2p+1 − 1 ln 2 −1 lim = lim p→∞ p→∞ p (p)0 ¡ p+1 ¢0 2 −1 = ln 2 = lim p→∞ 2p+1 − 1 and ¸ · ln (p + 1) lim = 0, p→∞ p consequently lim q = 2. p→∞
Taking the limit over p → ∞ in (2.7), we deduce b ≤ 1 + 2Db,
for b > 1
from where we get b−1 , b > 1. 2b Taking the limit over b → ∞ in (2.8) we conclude that D ≥ 12 , showing that the constant 12 in (2.5) cannot be replaced by a smaller quantity in (2.5). ¤ (2.8)
D≥
3. Some Inequalities of Landau Type on Unbounded Intervals The following technical lemma will be used in the following (see also [4]). Lemma 1. Let C, D > 0 and r, u ∈ (0, 1]. Consider the function gr,u : (0, ∞) → (0, ∞) given by C (3.1) gr,u (λ) = u + Dλr . λ Define µ ¶ 1 uC r+u (3.2) λ0 := ∈ (0, ∞) . rD Then we have (3.3)
inf
λ∈(0,∞)
gr,u (λ) = g (λ0 ) =
r+u u
u r+u
·r
u r+u
r
r
C r+u D r+u .
Proof. We observe that rDλr+u − Cu , λ ∈ (0, ∞) . λu+1 0 (λ) = 0, λ ∈ (0, ∞) is λ provided The unique solution of the equation gr,u 0 by (3.2). 0 gr,u (λ) =
28
S.S. DRAGOMIR
The function gr,u is decreasing on (0, λ0 ) and increasing on (λ0 , ∞) . The global minimum for gr,u on (0, ∞) is µ ¶ r C uC r+u gr,u (λ0 ) = ¡ ¢ u + D uC r+u rD rD
=
u and the equality (3.3) is proved.
r+u
u r+u
r
r r+u
r
u
C r+u D r+u ¤
The following particular cases are useful in applications. Corollary 1. Let C, D > 0. (i) For r ∈ (0, 1], consider the function gr : (0, ∞) → (0, ∞) , given by C (3.4) gr (λ) = + Dλr . λ Define µ ¶ 1 C r+1 λ0 = ∈ (0, ∞) . (3.5) rD Then we have ¡ ¢ r+1 r 1 (3.6) inf gr (λ) = gr λ0 = C r+1 D r+1 . r λ∈(0,∞) r r+u (ii) For u ∈ (0, 1], consider the function gu : (0, ∞) → (0, ∞) , given by C (3.7) gu (λ) = u + Dλ. λ Define ¶ 1 µ uC 1+u f ∈ (0, ∞) . λ0 = D Then we have ³ ´ 1+u 1 u f0 = u+1 D u+1 . inf gu (λ) = gu λ u C λ∈(0,∞) u 1+u The following result holds. Theorem 3. Let J be an unbounded subinterval of R and g : J → R a locally absolutely continuous function on J. If g ∈ L∞ (J), the derivative g 0 : J → R is of locally bounded variation and there exists a constant VJ > 0 and r ∈ (0, 1] such that ¯ ¯ ¯_b ¡ 0 ¢¯ r ¯ ¯ for any a, b ∈ J; (3.8) ¯ a g ¯ ≤ VJ |a − b| then g 0 ∈ L∞ (J) and one has the inequality r
(3.9)
1 r ° 0° 2 r+1 (r + 1) r+1 r+1 °g ° ≤ V . kgk r J,∞ J J,∞ r r+1
INEQUALITIES FOR FUNCTIONS OF BOUNDED VARIATION
29
Proof. Applying Theorem 1 for the function f = g 0 on [a, b] (or [b, a]), we deduce ¯ ¯ ° 0° |g (b) − g (a)| ¯¯_b ¡ 0 ¢¯¯ ° ° (3.10) +¯ g ¯. g [a,b],∞ ≤ a |b − a| for any a, b ∈ J, a 6= b. Since |g 0 (b)| ≤ kg 0 k[a,b],∞ , |g (b) − g (a)| ≤ 2 kgkJ,∞ , then by (3.8) and (3.10) we deduce (3.11)
¯ 0 ¯ 2 kgkJ,∞ ¯g (b)¯ ≤ + VJ |b − a|r |b − a|
for any a, b ∈ J, a 6= b. Fix b ∈ J. Then for any λ > 0, there exists an a ∈ J such that λ = |b − a| . Consequently, by (3.11), we deduce that (3.12)
¯ 0 ¯ 2 kgkJ,∞ ¯g (b)¯ ≤ + VJ λr λ
for any λ > 0 and b ∈ J. Taking the infimum over λ ∈ (0, ∞) in (3.12) and using Corollary 1, we deduce ´ r 1 ¯ 0 ¯ r+1³ r+1 ¯ ¯ (3.13) g (b) ≤ 2 kgkJ,∞ · VJr+1 r r r+1 r 1 r 2 r+1 (r + 1) r+1 = VJr+1 kgkJ,∞ r r r+1 for any b ∈ J. Finally, taking the supremum in (3.13) over b ∈ J, we deduce the desired result (3.9). ¤ There are a number of particular cases of interest. Corollary 2. Assume that g : J → R is such that g 0 : J → R is locally absolutely continuous and g 00 ∈ L∞ (J) . If g ∈ L∞ (J) , then g 0 ∈ L∞ (J) and 1 √ ° 0° ° 00 ° 1 2 °g ° °g ° 2 . (3.14) ≤ 2 2 kgk J,∞ J,∞ J,∞ Proof. If g 00 ∈ L∞ (J) , then ¯ ¯_ ¯ ¯Z ° ° ¯ b ¡ 0 ¢¯ ¯ b ¯ 00 ¯ ¯ ¯ ¯ ¯ ≤ |b − a| °g 00 ° ¯ ¯ ¯ g (t) dt = g ¯ ¯ ¯ ¯ a J,∞ a
for any a, b ∈ J, giving, by (3.11), that (3.15)
¯ 0 ¯ 2 kgkJ,∞ ° 00 ° ¯g (b)¯ ≤ + °g °J,∞ |b − a| |b − a|
for any a, b ∈ J, a 6= b. Applying Theorem 3 for VJ = kg 00 kJ,∞ and r = 1, we deduce (3.14). The following result is also of interest.
¤
30
S.S. DRAGOMIR
Corollary 3. Assume that g : J → R is such that g 0 ∈ Lp (J) , p > 1. If g ∈ L∞ (J) , then g 0 ∈ L∞ (J) and (3.16)
° 0° °g ° ≤ J,∞
p−1
2 2p−1 (2p − 1) p−1
p
(p − 1) 2p−1 p 2p−1
p−1 ° ° p−1 2p−1 ° 00 ° 2p−1 · kgkJ,∞ g J,p .
Proof. Using H¨older’s inequality, we have ¯ ¯Z ¯1 ¯ ¯ ¯Z ¯_b ¡ 0 ¢¯ ¯ b ¯ 00 ¯ ¯ ¯ b ¯ q ¯g (t)¯ dt¯ ≤ ¯ ¯ ¯ ¯ dt¯¯ ¯ ¯ ¯ a g ¯=¯ a a ° 1 ° ≤ |b − a| q °g 00 °J,p , p > 1,
¯Z b ¯1 ¯ ¯ 00 ¯p ¯ p ¯ ¯g (t)¯ dt¯ ¯ ¯ a
1 1 + =1 p q
for any a, b ∈ J, giving, by (3.11), that (3.17)
° ¯ 0 ¯ 2 kgkJ,∞ 1 ° ¯g (b)¯ ≤ + |b − a| q °g 00 °J,p , |b − a|
for any a, b ∈ J, a 6= b. Applying Theorem 3 for VJ = kg 00 kJ,p and r = (3.16).
1 q
=
p−1 p ,
we deduce ¤
The following result also holds. Theorem 4. Let J be an unbounded subinterval of R and g : J→ R a locally absolutely continuous function on J. If g 0 ∈ L1 (J) , the derivative g 0 : J → R is of locally bounded variation and there exists a constant VJ > 0 and r ∈ (0, 1] such that ¯ ¯ ¯_b ¡ 0 ¢¯ r ¯ ¯ (3.18) for any a, b ∈ J; ¯ a g ¯ ≤ VJ |a − b| then g 0 ∈ L∞ (J) and one has the inequality 1 ° 0° ° r r+1° °g ° °g 0 ° r+1 V r+1 . (3.19) ≤ r J,1 J J,∞ r r+1 Proof. Since, for any a, b ∈ J, ¯Z b ¯ ¯Z b ¯ ¯ ¯ ¯ ¯ 0 ¯ ¯ ° 0° 0 ¯ ¯ ¯ ¯ ¯ |g (b) − g (a)| ≤ ¯ g (s) ds¯ ≤ ¯ g (s) ds¯¯ ≤ °g °J,1 , a
a
then, by (3.10) and (3.18), we deduce ¯ 0 ¯ kg 0 kJ,1 ¯g (b)¯ ≤ + VJ |b − a|r |b − a| for any a, b ∈ J, a 6= b. Using an argument similar to the one in Theorem 3, we deduce (3.19). ¤ The following particular case also holds.
INEQUALITIES FOR FUNCTIONS OF BOUNDED VARIATION
31
Corollary 4. Assume that g : J → R is such that g 0 : J → R is locally absolutely continuous and g 00 ∈ L∞ (J) . If g 0 ∈ L1 (J) , then g 0 ∈ L∞ (J) and ° 0° ° 0 ° 1 ° 00 ° 1 °g ° °g ° 2 °g ° 2 . (3.20) ≤ 2 J,∞ J,1 J,∞ Corollary 5. Assume that g : J → R is such that g 0 : J → R is locally absolutely continuous and g 00 ∈ Lp (J) , p > 1. If g 0 ∈ L1 (J) , then g 0 ∈ L∞ (J) and (3.21)
° 0° °g ° ≤ J,∞
2p − 1 (p − 1)
p−1 2p−1
p
p 2p−1
p−1 ° ° p−1 2p−1 ° 00 ° 2p−1 · kgkJ,1 g J,p .
We may state the following result as well. Theorem 5. Let J be an unbounded subinterval of R and g : J → R a locally absolutely continuous function on J. If g 0 ∈ Lα (J) , α > 1, the derivative g 0 : J → R is of locally bounded variation on J and there exists a constant VJ > 0 and r ∈ (0, 1] such that ¯_ ¯ ¯ b ¡ 0 ¢¯ r ¯ ¯ (3.22) for any a, b ∈ J; ¯ a g ¯ ≤ VJ |b − a| then g 0 ∈ L∞ (J) and one has the inequality 1 ° 0° ° αr αr + 1 ° °g 0 ° αr+1 V αr+1 . °g ° (3.23) ≤ αr αr J,α J J,∞ α αr+1 r αr+1 Proof. By H¨older’s integral inequality, we have ¯Z b ¯ ¯Z b ¯ ¯ ¯ ¯ ¯ 0 ¯ ¯ 0 ¯ ¯ ¯ ¯ ¯ |g (b) − g (a)| = ¯ g (s) ds¯ ≤ ¯ g (s) ds¯¯ a
° ° ≤ |b − a| °g 0 °J,α , 1 β
a
α > 1,
1 1 + = 1, α β
and then, by (3.10) and (3.18), we deduce 1
(3.24)
¯ 0 ¯ |b − a| β kg 0 kJ,α ¯g (b)¯ ≤ + |b − a|r VJ |b − a| kg 0 kJ,α r = 1 + |b − a| VJ α |b − a|
for any a, b ∈ J, a 6= b. Fix b ∈ J. Then for any λ > 0, there exists an a ∈ J such that λ = |b − a| . Consequently, by (3.14) we deduce that (3.25) for any λ > 0 and b ∈ J.
¯ 0 ¯ kg 0 kJ,α ¯g (b)¯ ≤ + λr VJ 1 λα
32
S.S. DRAGOMIR
Taking the infimum over λ ∈ (0, ∞) in (3.25) and using Lemma 1 for u = α1 , we deduce ¯ 0 ¯ ¯g (b)¯ ≤
r+ 1
¡ 1 ¢ r+α 1
1
° 0 ° r+r 1 r+α 1 °g ° α V α J,α J
1 α r
1
r r+ α 1 ° αr αr + 1 ° αr+1 0 ° αr+1 ° g = V αr αr J,α J α αr+1 r αr+1 α
α
for any b ∈ J, giving the desired result (3.23).
¤
The following corollary holds. Corollary 6. Assume that g : J → R is such that g 0 is locally absolutely continuous and g 00 ∈ L∞ (J) . If g 0 ∈ Lα (J) , α > 1, then g 0 ∈ L∞ (J) and (3.26)
° 0° ° α ° ° 1 α+1 ° °g ° °g 0 ° α+1 °g 00 ° α+1 . ≤ α J,α J,∞ J,∞ α α+1 r
Finally we have Corollary 7. Assume that g : J → R is such that g 0 is locally absolutely continuous and g 00 ∈ Lp (J) , p > 1. If g 0 ∈ Lα (J) , α > 1, then g 0 ∈ L∞ (J) and (3.27) α(p−1) p ° 0° ° 0 ° α(p−1)+p ° 00 ° α(p−1)+p α (p − 1) + p °g ° °g ° °g ° . ≤ α(p−1) α(p−1) J,α J,p p J,∞ α α(p−1)+p (p − 1) α(p−1)+p · p α(p−1)+p References [1] Z. Ditzian, Remarks, questions and conjectures on Landau-Kolmogorov-type inequalities, Math. Ineq. Appl., 3 (2000), 15-24. [2] S.S. Dragomir, The Ostrowski integral inequality for mappings of bounded variation, Bull. Austral. Math. Soc., 60 (1999), 145-156. [3] S.S. Dragomir, On the Ostrowski’s integral inequality for mappings with bounded variation and applications, Math. Ineq. & Appl., 4(1) (2001), 59-66. [4] S.S. Dragomir and C.I. Preda, Some Landau type inequalities for functions whose derivatives are H¨ older continuous, Non. Anal. Forum (Korea), 9(1)(2004), 25-31. [5] G.H. Hardy and J.E. Littlewood, Some integral inequalities connected with the calculus of variations, Quart. J. Math. Oxford Ser., 3 (1932), 241-252. [6] G.H. Hardy, E. Landau and J.E. Littlewood, Some inequalities satisfied by the integrals or derivatives of real or analytic functions, Math. Z., 39 (1935), 677-695. 00 [7] R.R. Kallman and G.-C. Rota, On the inequality kf 0 k2 ≤ 4kf k · kf k, in “Inequalities”, Vol. II (O. Shisha, Ed) pp. 187-192. Academic Press, New York, 1970. ur zweimal differentzierban funktionen, Proc. Lon[8] E. Landau, Einige Ungleichungen f¨ don Math. Soc., 13 (1913), 43-49. J. E. Peˇcari´c, On Landau type inequalities [9] L. Maranguni´c and, for functions with H¨ older continuous derivatives, Journal of Inequl. Pure & Appl. Math., 5(2004), Issue 3, Article 72 [Online: http://jipam.vu.edu.au/user.php?op=displayuser&uid=67].
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[10] D.S. Mitrinovi´c, J.E. Peˇcari´c and A.M. Fink, Inequalities Involving Functions and their Integrals and Derivatives, Kluwer Academic Publishers, Dordrecht/Boston/London, 1991. [11] C.P. Niculescu and C. Bu¸se, The Hardy-Landau-Littlewood inequalities with less smoothness, J. Inequal. in Pure and Appl. Math., 4(2003), No. 3, Article 51 [Online: http://jipam.vu.edu.au/article.php?sid=289]. School of Computer Science and Mathematics, Victoria University of Technology, PO Box 14428, Melbourne VIC 8001, Australia. E-mail address:
[email protected] URL: http://rgmia.vu.edu.au/SSDragomirWeb.html
