A Hardy inequality and applications to reverse Holder inequalities for weights on R
aa r X i v : . [ m a t h . F A ] D ec A HARDY INEQUALITY AND APPLICATIONS TO REVERSEH ¨OLDER INEQUALITIES FOR WEIGHTS ON R ELEFTHERIOS N. NIKOLIDAKIS
Abstract.
We prove a sharp integral inequality valid for non-negative functions definedon [0 , L norm. This is in fact a generalization of the well known integralHardy inequality. We prove it as a consequence of the respective weighted discreteanalogue inequality which proof is presented in this paper. As an application we findthe exact best possible range of p > q such that any non-increasing g which satisfiesa reverse H¨older inequality with exponent q and constant c upon the subintervals of(0 , p and adifferent in general constant c ′ . The result has been treated in [1] but here we give analternative proof based on the above mentioned inequality.1. Introduction
During his efforts to simplify the proof of Hilbert’s double series theorem, G. H.Hardy [7], first proved in 1920 the most famous inequality which is known in theliterature as Hardy’s inequality (see also [10], Theorem 3.5). This is stated as
Theorem A. If p > , a n > , and A n = a + a + · · · + a n , n ∈ N , then ∞ X n =1 (cid:18) A n n (cid:19) p < (cid:18) pp − (cid:19) p ∞ X n =1 a pn . (1.1) Moreover, inequality (1.1) is best possible, that is the constant and the right side cannotbe decreased .In 1926, E.Copson, generalized in [3] Theorem A by replacing the arithmetic meanof a sequence by a weighted arithmetic mean. More precisely he proved the following
Theorem B.
Let p > , a n , λ n > , for n = 1 , , . . . . Further suppose that Λ n = n P i =1 λ i and A n = n P i =1 λ i a i . Then ∞ X n =1 λ n (cid:18) A n Λ n (cid:19) p ≤ (cid:18) pp − (cid:19) p ∞ X n =1 λ n a pn , (1.2) where the constant involved in (1.2) is best possible. In [3], Copson proves also a second weighted inequality, which as Hardy noted in[8], can be derived from Theorem B. From then and until now there have been given Mathematics Subject Classification.
Primary 26D15; Secondary 42B25. Keywords and phrases.
Hardy inequalities, Reverse H¨older inequalities, weights. several generalizations of the above two inequalities. The first one is given by Hardyand Littlewood who generalized in a specific direction Theorem 1.2 (see [9]). This wasgeneralized further by Leindler in [14], and by Nemeth in [17]. Also in [16] one can seefurther generalizations of Hardy’s and Copson’s series inequalities by replacing meansby more general linear transforms. For the study of Copson’s inequality one can also see[4]. Additionally, in [5], Elliot has already proved inequality (1.2) by similar methodsto those that appear in [3].There is a continued analogue of Theorem 1.1 (see [10]) which can be stated as
Theorem C. If p > , f ( x ) ≥ for x ∈ [0 , + ∞ ) then Z ∞ (cid:18) x Z x f ( t ) dt (cid:19) p dx < (cid:18) pp − (cid:19) p Z ∞ f p ( x ) dx, (1.3)Further generalizations of (1.3) can be seen in [8]. Other authors have also studiedthese inequalities in more general forms as it may be seen in [15] and [20]. E. Landauhas also studied the above inequality and his work appears in [13]. For a completediscussion of the topic one can consult [12] and [19]. In this paper we generalize (1.3)by proving the following Theorem 1.
Let g : [0 , → R + be integrable function, p > , and additionallyassume that R g = f . Then the following inequality is true, for any q such that ≤ q ≤ p Z (cid:18) t Z t g (cid:19) p dt < (cid:18) pp − (cid:19) q Z (cid:18) t Z t g (cid:19) p − q g q ( t ) dt − qp − f p (1.4) Moreover, inequality (1.4) is sharp in the sense that, the constant ( pp − ) q cannot bedecreased, while the constant qp − cannot be increased for any fixed f . In fact we are going to prove, an even more general inequality which is the discreteanalogue of (1.4) for the case q = 1, which is weighted. This is a generalization of (1.2)and is described in the following Theorem 2.
Let ( a n ) n be a sequence of non-negative real numbers. We define for everysequence ( λ n ) n of positive numbers the following quantities A n = λ a + · · · + λ n a n and Λ n = λ + · · · + λ n . Then the following inequality is true: N X n =1 λ n (cid:18) A n Λ n (cid:19) p ≤ (cid:18) pp − (cid:19) N X n =1 λ n a n (cid:18) A n Λ n (cid:19) p − − p − Λ N (cid:18) A N Λ N (cid:19) p , (1.5) for any N ∈ N . It is obvious that by setting λ n = 1 for every n ∈ N , in Theorem 2, we reach, for q = 1, to the discrete analogue of (1.4), thus generalizing (1.1) and (1.2). Then Theorem1 is an easy consequence, for the case q = 1, by the use of a standard approximationargument, of L functions on (0 , HARDY TYPE INEQUALITY 3 q ∈ [1 , p ]. We mention also that the opposite problem for negative exponents is treatedin [18].We believe that Theorem 1 has many applications in many fields and especially in thetheory of weights. Our intention in this paper is to describe one of them. We mentionthe related details. Let Q ⊆ R N be a given cube. Let also p > h : Q → R + be such that h ∈ L p ( Q ). Then, as it is well known, the following, named as H¨older’sinequality is satisfied (cid:18) | Q | Z Q h (cid:19) p ≤ | Q | Z Q h p , for any cube Q ⊆ Q . In this paper we are interested for functions that satisfy a reverse H¨older inequality.More precisely we say that h satisfies the reverse H¨older inequality with exponent q > c ≥ | Q | Z Q h q ≤ c · (cid:18) | Q | Z Q h (cid:19) q for every cube Q ⊆ Q . (1.6)Now in [6] it is proved the following. Theorem A.
Let < q < ∞ and h : Q → R + such that (1.6) holds. Then thereexists ε = ε ( N, q, c ) such that h ∈ L p for any p such that p ∈ [ q, q + ε ) . Moreover thefollowing inequality holds | Q | Z Q h p ≤ c ′ (cid:18) | Q | Z Q h (cid:19) p , for any cube Q ⊆ Q , p ∈ [ q, q + ε ) and some constant c ′ = c ′ ( N, p, q, c ).As a consequence the following question naturally arises and is posed in [2] . Whatis the best possible value of ε ? The problem for the case N = 1 was solved in [1]for non-increasing functions g and was completed for arbitrary functions in [11]. Moreprecisely in [1] it is shown the following Theorem B.
Let g : (0 , → R + be non-increasing which satisfies the following in-equality b − a Z ba g q ≤ c (cid:18) b − a Z ba g (cid:19) q , (1.7) for every ( a, b ) ⊆ (0 , , where q > is fixed, and c independent of a, b . If we define p > q as the root of the following equation p − qp · (cid:18) p p − (cid:19) q · c = 1 , (1.8) we have that g ∈ L p ((0 , and g satisfies a reverse H¨older inequality with exponent p ,for every p such that p ∈ [ q, p ) . Moreover the result is sharp, that is the value of p cannot be increased. ELEFTHERIOS N. NIKOLIDAKIS
The problem was solved completely in [11] where the notion of the non-increasingrearrangement of h was used and which is defined as follows: h ∗ ( t ) = sup e ⊆ (0 , | e |≥ t h inf x ∈ e h ( x ) i . More precisely the following appears in [11].
Theorem C.
Let h : (0 , → R + , that it satisfies (1.7), for every ( a, b ) ⊆ [0 , .with q > fixed and c ≥ . Then the same inequality is true if we replace h by it’snon-increasing rearrangement .It is immediate now that Theorem B and C answer the question as it was posed in[2], for the case N = 1.Our aim in this paper is to give an alternative proof of Theorem B by using Theorem1. We will prove the following variant of Theorem B which we state as Theorem 3.
Let g : (0 , → R + be non-increasing satisfying a reverse H¨older inequal-ity with exponent q > and constant c ≥ upon all intervals of the form (0 , t ] . Thatis the following hold: t Z t g q ≤ c · (cid:18) t Z t g (cid:19) q , (1.9) for any t ∈ (0 , . Then for every p ∈ [ q, p ) the following inequality true t Z t g p ≤ c ′ (cid:18) t Z t g (cid:19) p , (1.10) for any t ∈ (0 , where c ′ = c ′ ( p, q, c ) and p is defined by (1.5). As a consequence g ∈ L p for every p ∈ [ q, p ).By the same reasoning we can prove the analogue of Theorem 3, for intervals of theform ( t, Theorem D.
Let g : (0 , → R + be non-increasing. Then (1.7) is satisfied for allsubintervals of (0 , iff it is satisfied for all subintervals of the form (0 , t ] and [ t, t, The Hardy type inequality
We first present the following which can be seen in [3].
Proof of Theorem 2.
For each n ∈ N define ∆ n = λ n (cid:18) A n Λ n (cid:19) p − pp − λ n (cid:18) A n Λ n (cid:19) p − a n = λ n ∆ ′ n , where ∆ ′ n = (cid:18) A n Λ n (cid:19) p − pp − (cid:18) A n Λ n (cid:19) p − a n . HARDY TYPE INEQUALITY 5
Obviously, a n = A n − A n − λ n for every n ∈ N , so we have ∆ ′ n = (cid:18) A n Λ n (cid:19) p − pp − (cid:18) A n Λ n (cid:19) p − A n − A n − λ n = (cid:18) A n Λ n (cid:19) p − pp − (cid:18) A n Λ n (cid:19) p Λ n λ n + pp − (cid:18) A n Λ n (cid:19) p − A n − λ n = (cid:18) A n Λ n (cid:19) p (cid:20) − pp − · Λ n λ n (cid:21) + 1 p − (cid:26) p · (cid:18) A n Λ n (cid:19) p − A n − Λ n − (cid:27) Λ n − λ n . (2.11)We now use the following elementary inequality px p − y ≤ ( p − x p + y p , which holds for any p > x, y ≥ x = A n Λ n , y = A n − Λ n − , so using (2.11) we have that: ∆ ′ n ≤ (cid:18) A n Λ n (cid:19) p (cid:20) − pp − Λ n λ n (cid:21) + 1 p − (cid:20) ( p − (cid:18) A n Λ n (cid:19) p + (cid:18) A n − Λ n − (cid:19) p (cid:21) · Λ n − λ n = (cid:18) A n Λ n (cid:19) p (cid:20) − p − p Λ n λ n + Λ n − λ n (cid:21) + 1 p − (cid:18) A n − Λ n − (cid:19) p Λ n − λ n = − p − · Λ n λ n (cid:18) A n Λ n (cid:19) p + 1 p − Λ n − λ n (cid:18) A n − Λ n − (cid:19) p . (2.12)Thus from (2.12) and the definition of ∆ n we conclude ∆ n ≤ p − Λ n − (cid:18) A n − Λ n − (cid:19) p − p − Λ n (cid:18) A n Λ n (cid:19) p , (2.13)This holds for every n ∈ N , n ≥ n = 1 we have the following equality ∆ = − p − Λ (cid:18) A Λ (cid:19) p . (2.14)For any N ∈ N we sum (2.13) from n = 2 to N and add also the equality (2.14), so weconclude after making the appropriate cancellations, inequality (1.5) of Theorem 2. (cid:3) The following is an easy consequence of the above result
Corollary 1:
Let g : [0 , → R + be integrable function, p > and additionallyassume that R g = f . Then the following inequality is true Z (cid:18) t Z t g (cid:19) p dt ≤ (cid:18) pp − (cid:19) Z (cid:18) t Z t g (cid:19) p − g ( t ) dt − p − f p . (2.15)We proceed now to the Proof of Theorem 1.
For any s ∈ [0 , p ] we define by I s by I s = Z (cid:18) t Z t g (cid:19) p − s g s ( t ) dt, ELEFTHERIOS N. NIKOLIDAKIS for any g : [0 , → R + integrable function, such that R g = f . Then, for the proof ofinequality (1.4), we just need to prove that I ≤ (cid:18) pp − (cid:19) q I q − qp − f p , for any q ∈ (1 , p ].We write I = Z g ( t ) (cid:18) t Z t g (cid:19) ( p − q ) /q (cid:18) t Z t g (cid:19) p − pq dt. We then apply in the above integral H¨older’s inequality, with exponents q, qq − , andwe have as a consequence that I ≤ I /qq I ( q − /q . (2.16)Additionally from Corollary 1 we obtain I ≤ pp − I − p − f p . (2.17)We consider now the difference L q = I − ( pp − ) q I q . We need to prove that L q ≤ − qp − f p .By using the inequalities (2.16) and (2.17) we have that L q ≤ I − (cid:18) pp − (cid:19) q I q I q − ≤ I − (cid:18) pp − (cid:19) q I − q +10 (cid:18) p − p I + 1 p f p (cid:19) q . (2.18)We define now the following function of the variable x > G ( x ) = x − (cid:18) pp − (cid:19) q x − q +1 (cid:18) p − p x + 1 p f p (cid:19) q . Then G ( x ) = x − x − q +1 (cid:18) x + 1 p − f p (cid:19) q , so that G ′ ( x ) = 1 + ( q − (cid:18) f p ( p − x (cid:19) q − q (cid:18) f p ( p − x (cid:19) q − . Now we consider the following function of the variable t ≥ F ( t ) = 1+( q − t q − qt q − .Then F ′ ( t ) = q ( q − t q − ( t − >
0, for every t >
1. Thus F is strictly increasingon its domain, so that F ( t ) > F (1) = 0, for any t >
1. We immediately conclude that G ′ ( x ) >
0, for every x >
0. As a consequence G is strictly increasing on (0 , + ∞ ). Weevaluate now lim x → + ∞ G ( x ) = l . We have that l = lim x → + ∞ x (cid:20) − (cid:18) f p ( p − x (cid:19) q (cid:21) = lim x → + ∞ − (cid:18) yf p p − (cid:19) q y = − qp − f p , (2.19) HARDY TYPE INEQUALITY 7 by using De’l Hospital’s rule. Thus since G is strictly increasing on (0 , + ∞ ), we havethat G ( x ) < − qp − f p , for any x >
0. Thus (2.18) yields L q < − qp − f p , which isinequality (1.4). We now prove its sharpness.We let J ′ = Z (cid:18) t Z t g (cid:19) p dt, and J ′ q = Z (cid:18) t Z t g (cid:19) p − q g q ( t ) dt for any 1 ≤ q ≤ p. Let also g = g a , where g a is defined for any a ∈ (0 , /p ), by g a ( t ) = t − a , t ∈ (0 , t ∈ (0 ,
1] we have that1 t Z t g a = 11 − a g a ( t ) and so J ′ J ′ q = (cid:16) − a (cid:17) p Z g pa dt (cid:16) − a (cid:17) p − q Z g pa dt = (cid:18) − a (cid:19) q . (2.20)Letting a → /p − in (2.20) we obtain that the constant ( pp − ) q , on the right of inequality(1.4), cannot be decreased. We now prove the second part of the sharpness of Theorem1. For this purpose we define for any fixed f >
0, and any a ∈ (0 , /p ), the function g a ( t ) = f (1 − a ) t − a , for every t ∈ (0 , R g a = f , t R t g a ( u ) du = − a g a ( t ), and that R g pa = f p (1 − a ) p − ap . We consider now the difference L q ( a ) = Z (cid:18) t Z t g a (cid:19) p dt − (cid:18) pp − (cid:19) q Z (cid:18) t Z t g a (cid:19) p − q g qa ( t ) dt = (cid:18) − a (cid:19) p Z g pa − (cid:18) pp − (cid:19) q (cid:18) − a (cid:19) p − q Z g pa = (cid:18) − a (cid:19) p − q f p (cid:20)(cid:18) − a (cid:19) q − (cid:18) pp − (cid:19) q (cid:21) − ap . (2.21)Letting now a → /p − , we immediately see, by an application of De’l Hospital’s rulethat L q ( a ) → − qp − f p . We have just proved that the constant qp − , appearing in frontof f p , cannot be increased. That is, both constants appearing on the right of (1.4) arebest possible. (cid:3) Applications to reverse H¨older inequalities
We will need first a preliminary lemma which in fact holds under some additionalhypothesis for g even if it is not decreasing, which can be proved using integrationby parts. We present a version that we will need below which is proved by measureintegration techniques. More precisely we will prove the following Lemma 1.
Let g : (0 , → R + be a non-increasing function. Then the following ELEFTHERIOS N. NIKOLIDAKIS inequality is true for any p > and every δ ∈ (0 , Z δ (cid:18) t Z t g (cid:19) p dt = − p − (cid:18) Z δ g (cid:19) p δ p − + pp − Z δ (cid:18) t Z t g (cid:19) p − g ( t ) dt. (3.22) Proof.
By using Fubini’s theorem it is easy to see that Z δ (cid:18) t Z t g (cid:19) p dt = Z + ∞ λ =0 pλ p − (cid:12)(cid:12)(cid:12)(cid:12)(cid:26) t ∈ (0 , δ ] : 1 t Z t g ≥ λ (cid:27)(cid:12)(cid:12)(cid:12)(cid:12) dt. (3.23)Let now 1 δ Z δ g = f δ ≥ f = Z g . Then1 t Z t g > f δ , ∀ t ∈ (0 , δ ) while1 t Z t g ≤ f δ , ∀ t ∈ [ δ, . Let λ be such that: 0 < λ < f δ . Then for every t ∈ (0 , δ ] we take 1 t Z t g ≥ δ Z δ g = f δ > λ . Thus (cid:12)(cid:12)(cid:12)(cid:12)(cid:26) t ∈ (0 , δ ] : 1 t Z t g ≥ λ (cid:27)(cid:12)(cid:12)(cid:12)(cid:12) = | (0 , δ ] | = δ. Now for every λ > f δ there exists unique a ( λ ) ∈ (0 , δ ) such that 1 a ( λ ) Z a ( λ g = λ . It’sexistence is quaranteeded by the fact that λ > f δ , that g is non-increasing and that g (0 + ) = + ∞ which may without loss of generality be assumed (otherwise we work forthe λ ’s on the interval (0 , k g k ∞ ]). Then (cid:26) t ∈ (0 , δ ] : 1 t Z t g ≥ λ (cid:27) = (0 , a ( λ )] . Thus, from the above and (3.23) we conclude that Z δ (cid:18) t Z t g (cid:19) p dt = Z f δ λ =0 pλ p − · δ · dλ + Z + ∞ λ = f δ pλ p − a ( λ ) dλ = δ ( f δ ) p + Z + ∞ λ = f δ pλ p − λ (cid:18) Z a ( λ )0 g ( u ) du (cid:19) dλ (3.24) HARDY TYPE INEQUALITY 9 by the definition of a ( λ ). As a consequence ( ?? ) gives Z δ (cid:18) t Z t g (cid:19) p dt = 1 δ p − (cid:18) Z δ g (cid:19) p + Z + ∞ λ = f δ pλ p − (cid:18) Z a ( λ )0 g ( u ) du (cid:19) dλ = 1 δ p − (cid:18) Z δ g (cid:19) p + Z + ∞ λ = f δ pλ p − (cid:18) Z { u ∈ (0 ,δ ]:1 u R u g ≥ λ } g (cid:19) dλ = 1 δ p − (cid:18) Z δ g (cid:19) p + pp − Z δ g ( t ) h λ p − i t R t gλ = f δ dt = 1 δ p − (cid:18) Z δ g (cid:19) p + pp − (cid:20) Z δ (cid:18) t Z t g (cid:19) p − g ( t ) − (cid:18) Z δ g ( t ) dt (cid:19) f p − δ (cid:21) = − p − δ p − (cid:18) Z δ g (cid:19) p + pp − Z δ (cid:18) t Z t g (cid:19) p − g ( t ) dt, where in the third equality we have used Fubini’s theorem and the fact that 1 δ Z δ g = f δ .In this way we derived (3.22). (cid:3) We are now able to give the
Proof of Theorem 3.
Suppose we are given g : (0 , → R + non-increasing and δ ∈ (0 , g is (1.10) or that:1 t Z t g q ≤ c · (cid:18) t Z t g (cid:19) q , for every t ∈ (0 , . Let now p > q and set a = p/q > g q in place of g and a in that of p . We conclude that: Z δ (cid:18) t Z t g q (cid:19) p/q dt ≤ − qp − q δ p/q − (cid:18) Z δ g q (cid:19) p/q + pp − q Z δ (cid:18) t Z t g q (cid:19) p/q − g q ( t ) dt ⇒ δ Z δ (cid:20)(cid:18) t Z t g (cid:19) p/q − g q ( t ) − p − qp (cid:18) t Z t g q (cid:19) p/q (cid:21) dt ≤ qp (cid:18) δ Z δ g q (cid:19) p/q . (3.25)Define now for every y > φ y with variable x by φ y ( x ) = x p/q − y − p − qp x p/q , for x ≥ y .Then φ ′ y ( x ) = ( p/q − x p/q − y − ( p/q − x p/q − = ( p/q − x p/q − ( y − x ) ≤ , for x ≥ y. Thus y ≤ x ≤ z ⇒ φ y ( x ) ≥ φ y ( z )(3.26)Let us now set in (3.26) x = 1 t Z t g q , y = g q ( t ) , z = c (cid:18) t Z t g (cid:19) q for any t ∈ (0 , . Then y ≤ x ≤ z ⇒ (cid:18) t Z t g q (cid:19) p/q − g q ( t ) − p − qp (cid:18) t Z t g q (cid:19) p/q ≥ c p/q − (cid:18) t Z t g (cid:19) p − q g q ( t ) − p − qp c p/q (cid:18) t Z t g (cid:19) p , ∀ t ∈ (0 , . As a consequence (3.25) gives, by using the last inequality the following1 δ Z δ (cid:18) t Z t g (cid:19) p − q g q ( t ) dt ≤ c · p − qp · δ Z δ (cid:18) t Z t g (cid:19) p dt + qp c (cid:18) δ Z δ g (cid:19) p , We use now the inequality,1 δ Z δ (cid:18) t Z t g (cid:19) p dt ≤ (cid:18) pp − (cid:19) q δ Z δ (cid:18) t Z t g (cid:19) p − q g q ( t ) dt which is a consequence of Theorem 1.We conclude that if p is defined by (1.8), for any p ∈ [ q, p ), the following holds (cid:20) − c p − qp (cid:18) pp − (cid:19) q (cid:21) δ Z δ (cid:18) t Z t g (cid:19) p − q g a ( t ) dt ≤ qp c (cid:18) δ Z δ g (cid:19) p , where 1 − c p − qp (cid:16) pp − (cid:17) q = k p >
0, for every such p . This becomes1 δ Z δ (cid:18) t Z t g (cid:19) p − q g q ( t ) dt ≤ q · cp · k p (cid:18) δ Z δ g (cid:19) p , (3.27)for any δ ∈ (0 , p ∈ [ q, p ).On the other hand 1 t Z t g ≥ g ( t ), since g is non-increasing, thus (3.27) ⇒ δ Z δ g p ≤ q · cp · k p (cid:18) δ Z δ g (cid:19) p , for any δ ∈ (0 ,
1] and p such that q ≤ p < p , which is an inequality of the form of(1.10), for suitable c ′ > c ≥ q > g a ( t ) = t − a where a = 1 /p , where p is defined by (1.5). Then it is easy to see that1 t Z t g q = c (cid:16) t Z t g (cid:17) q , for every t ∈ (0 , g / ∈ L p ((0 , p cannot be increased and Theorem 2 is proved. (cid:3) References [1] L. D’ Appuzzo and C. Sbordone,
Reverse H¨older inequalities. A sharp result . Rendiconti Math., , Ser. VII, (1990), 357-366.[2] B. Bojarski, Remarks on the stability of reverse H¨older inequalities and quasiconformal mappings ,Ann. Acad. Sci. Fenn. A Math. , (1985), 291-296.[3] E. T. Copson, Note on series of positive terms , J. London Math. Soc. (1927), 9-12 and (1928),49-51. HARDY TYPE INEQUALITY 11 [4] E. T. Copson,
Some integral inequalities , Proc. Roy. Soc. Edinburgh Sect. A (1975/1976),157-164.[5] E. B. Elliot, A simple exposition of some recently proved facts as to convergency . J. London Math.Soc., , (1926), 93-96.[6] F. W. Gehring, The L p inequalities of the partial derivatives of a quasiconformal mapping , ActaMath., , (1973), 265-27.[7] G. H. Hardy, Note on a theorem of Hilbert , Math Z. (1920), 314-317.[8] G. H. Hardy, Notes on some points in the integral calculus, L. X. An inequality between integrals . Messenger of Math. , (1928), 12-16.[9] G. H. Hardy and J. E. Littlewood, Elementary theorems concerning power series with positivecoefficients and moment constants of positive functions , J. Reine Angew. Math. (1927), 141-158.[10] G. H. Hardy, J. E. Littlewood and G. Polya,
Inequalities , Cambridge University Press, Cambridge(1934).[11] A. A. Korenovskii,
The exact continuation of a Reverse H¨older inequality and Muckenhoupt’sconditions , Math. Notes , (1992) 1192-1201.[12] A. Kufner, L. Maligranda and L. E. Persson, The prehistory of the Hardy inequality , Amer. Math.Monthly, Vol. , No 8, (2006), 715-732.[13] E. Landau,
A note on a theorem concerning series of positive terms: Extract from a letter of Prof.E. Landau to Prof. I. Schur , J. London Math. Soc. , (1926), 38-39.[14] L. Leindler, Generalization of inequalities of Hardy and Littlewood , Acta Sci. Math (1970),279-285.[15] N. Levinson, Generalizations of an inequality of Hardy , Duke Math. J. (1964), 389-394.[16] E. R. Love, Generalizations of Hardy’s and Copson’s inequalities , J. London Math. Soc. (1984),431-440.[17] J. Nemeth, Generalizations of the Hardy-Littlewood inequality , Acta Sci. Math. (Szeged) (1971),295-299.[18] E. N. Nikolidakis, A sharp integral Hardy type inequality and applications to Muckenhoupt weightson R , Ann. Acad. Sci. Fenn. Math. Vol 39, (2014), 887-896.[19] B. G. Pachpatte,
Mathematical Inequalities , North Holland Mathematical Library, Vol , (2005).[20] J. B. G. Pachpatte, On a new class of Hardy type inequalities , Proc. Roy. Soc. Edinburgh Sect. A (1987), 265-274.(1987), 265-274.