aa r X i v : . [ m a t h . C A ] A p r ON ALZER’S INEQUALITY
MOHAMMAD W. ALOMARI
Abstract.
Extensions and generalizations of Alzer’s inequality; which is ofWirtinger type are proved. As applications, sharp trapezoid type inequalityand sharp bound for the geometric mean are deduced. Introduction
In Fourier analysis, the theory of inequalities plays an important and useful rolein almost all branches of its analyzes. Early of the last century, several famousinequalities have been used in the theory of Fourier series, Fourier integrals andFourier transform. The inequalities of Bessel, Blaschke, Wirtinger, Beesack andothers, are used at large in convergence and estimations of such series and integrals.In [4], Wirtinger proved the following inequality regarding square integrable func-tions:
Theorem 1.
Let f be a real valued function with period π and R π f ( x ) dx = 0 .If f ′ ∈ L [0 , π ] , then Z π f ( x ) dx ≤ Z π f ′ ( x ) dx, (1.1) with equality if and only if f ( x ) = A cos x + B sin x , A, B ∈ R . Various generalizations, counterparts and refinements were considered in [1]–[6]and the references therein.In [1], Alzer introduced a Wirtinger like inequality for continuously differentiableperiodic functions, which reads:
Theorem 2. If f is a real valued continuously differentiable function with period π and R π f ( x ) dx = 0 , then π max ≤ x ≤ π f ( x ) ≤ Z π f ′ ( x ) dx. (1.2) Equality holds if and only if f ( x ) = c h (cid:0) x − ππ (cid:1) − i , ≤ x ≤ π and c ∈ R . The aim of this work is to extend and generalize Alzer inequality (1.2), by re-laxing the assumptions: continuity of f ′ , periodicity and the interval involved forvarious kind of functions. Date : November 14, 2018.2000
Mathematics Subject Classification.
Key words and phrases.
Wirtinger inequality, Alzer inequality, Trapezoid inequality. The Results
The version of Alzer inequality for convex functions may be stated as follows:
Theorem 3.
Let f : I ⊆ R → R be a convex mapping on I ◦ , the interior of theinterval I , where a, b ∈ I ◦ with a < b , such that f ′ ∈ L [ a, b ] . If f ( a ) f ( b ) > and R ba f ( t ) dt = 0 , then the inequality f ( a ) f ( b ) ≤ b − a · Z ba f ′ ( x ) dx, (2.1) holds. The constant ‘ b − a ’ is the best possible, in the sense that it cannot be replacedby a smaller constant.Proof. Assume that f attains its maximum value at x ∈ [ a, b ] and let max a ≤ x ≤ b f ( x ) = f ( x ), for some a ≤ x ≤ b , then0 ≤ Z ba " f ′ ( x ) f ( x ) − b − a ) · (cid:18) x − a + b (cid:19) dx = Z ba f ′ ( x ) f ( x ) dx − b − a ) f ( x ) Z ba (cid:18) x − a + b (cid:19) f ′ ( x ) dx (2.2) + 144( b − a ) Z ba (cid:18) x − a + b (cid:19) dx. Observing that Z ba (cid:18) x − a + b (cid:19) f ′ ( x ) dx = b − a · [ f ( a ) + f ( b )] − Z ba f ( x ) dx, taking in account that R ba f ( x ) dx = 0. Substituting in (2.2), we get0 ≤ Z ba " f ′ ( x ) f ( x ) − b − a ) · (cid:18) x − a + b (cid:19) dx = Z ba f ′ ( x ) f ( x ) dx − b − a ) f ( x ) [ f ( a ) + f ( b )] + 12( b − a )which gives that { f ( a ) + f ( b ) − f ( x ) } · max a ≤ x ≤ b f ( x ) ≤ b − a · Z ba f ′ ( x ) dx. Finally, since f is convex then f attains its maximum at the endpoints ‘ a ’ or ‘ b ’,so if max a ≤ x ≤ b f ( x ) = f ( b ) = f ( x ), thus we have f ( a ) · max a ≤ x ≤ b f ( x ) ≤ b − a · Z ba f ′ ( x ) dx, (2.3)and if max a ≤ x ≤ b f ( x ) = f ( a ) = f ( x ), we have f ( b ) · max a ≤ x ≤ b f ( x ) ≤ b − a · Z ba f ′ ( x ) dx. (2.4) N ALZER’S INEQUALITY 3
So that the both inequalities (2.3) and (2.4), can be read as f ( a ) f ( b ) ≤ b − a · Z ba f ′ ( x ) dx, and thus the proof of (2.11) is established. To prove the sharpness of (2.11), let(2.11) holds with another constant C > f ( a ) f ( b ) ≤ C · Z ba f ′ ( x ) dx. (2.5)Define the function f : [0 , → R defined by f ( x ) = 6 x − x + 1, for all x ∈ [0 , f is convex for all x ∈ [0 , f (0) = f (1) = 1, and R f ′ ( x ) dx = 12. Making use of (2.5), we have C ≥ , and this proves the bestpossibility of , which completes the proof. (cid:3) The following inequality for monotonic mappings holds.
Theorem 4.
Let f : I ⊆ R → R be an increasing function on I ◦ , the interior ofthe interval I , where a, b ∈ I ◦ with a < b , such that f ′ ∈ L [ a, b ] . If R ba f ( t ) dt = 0 ,then the inequality [2 f ( a ) − f ( b )] · f ( b ) ≤ b − a · Z ba f ′ ( x ) dx, (2.6) holds. The constant ‘ b − a ’ is the best possible.Proof. Repeating the steps in the proof of Theorem 7, since f is bounded andmonotonically increasing on [ a, b ], then f ( a ) ≤ f ( t ) for all t ∈ [ a, b ], therefore0 ≤ Z ba " f ′ ( x ) f ( x ) − b − a ) · (cid:18) x − a + b (cid:19) dx = Z ba f ′ ( x ) f ( x ) dx − b − a ) f ( x ) [ f ( a ) + f ( b )] + 12( b − a ) ≤ Z ba f ′ ( x ) f ( x ) dx − b − a ) f ( x ) f ( a ) + 12( b − a )which gives that [2 f ( a ) − f ( b )] · f ( b ) ≤ b − a · Z ba f ′ ( x ) dx, which proves the inequality (2.6). The sharpness holds with the function f ( x ) =4 c · x + 12 c · x − c − c , for all x ∈ [0 , c = √ . (cid:3) Corollary 1.
Let f : I ⊆ R → R be a bounded decreasing function on I ◦ , theinterior of the interval I , where a, b ∈ I ◦ with a < b , such that f ′ ∈ L [ a, b ] . If R ba f ( t ) dt = 0 , then the inequality [2 f ( b ) − f ( a )] · f ( a ) ≤ b − a · Z ba f ′ ( x ) dx, (2.7) holds. The constant ‘ b − a ’ is the best possible.Proof. The proof is similar the proof of Theorem 4. (cid:3)
M.W. ALOMARI
In general, we may generalize and extend Alzer inequality (1.2) as follows:
Theorem 5.
Let f : I ⊆ R → R be an absolutely continuous mapping on I ◦ , theinterior of the interval I , where a, b ∈ I ◦ with a < b , such that f ′ ∈ L [ a, b ] . If f ( a ) = max a ≤ x ≤ b f ( x ) = f ( b ) and R ba f ( t ) dt = 0 , then the inequality max a ≤ x ≤ b f ( x ) ≤ b − a · Z ba f ′ ( x ) dx, (2.8) holds. The constant ‘ b − a ’ is the best possible.Proof. Given the assumptions. Assume that f attains its maximum value at x ∈ [ a, b ] and let max a ≤ x ≤ b f ( x ) = f ( x ), for some a ≤ x ≤ b , then0 ≤ Z ba " f ′ ( x ) f ( x ) − b − a ) · (cid:18) x − a + b (cid:19) dx = Z ba f ′ ( x ) f ( x ) dx − b − a ) f ( x ) Z ba (cid:18) x − a + b (cid:19) f ′ ( x ) dx (2.9) + 144( b − a ) Z ba (cid:18) x − a + b (cid:19) dx. Since f ( a ) = max a ≤ x ≤ b f ( x ) = f ( b ), we have Z ba (cid:18) x − a + b (cid:19) f ′ ( x ) dx = b − a · [ f ( a ) + f ( b )] = ( b − a ) · f ( x )Substituting in (2.9),0 ≤ Z ba " f ′ ( x ) f ( x ) − b − a ) · (cid:18) x − a + b (cid:19) dx = Z ba f ′ ( x ) f ( x ) dx − b − a + 12 b − a = Z ba f ′ ( x ) f ( x ) dx − b − a which gives that max a ≤ x ≤ b f ( x ) ≤ b − a · Z ba f ′ ( x ) dx, and thus the proof of (2.9) is established. To prove the sharpness of (2.8), let a = 0, b = 2 π , then (2.8) reduces to (1.2), so by considering the same function f as givenin Theorem 2, we get the sharpness. (cid:3) The most extensive case holds without any additional restrictions on f is con-sidered as follows: Theorem 6.
Let f : I ⊆ R → R be an absolutely continuous mapping on I ◦ , theinterior of the interval I , where a, b ∈ I ◦ with a < b , such that f ′ ∈ L [ a, b ] . Thenthe inequality (cid:20) b − a · T rap ( f ) − max a ≤ x ≤ b f ( x ) (cid:21) · max a ≤ x ≤ b f ( x ) ≤ b − a · Z ba f ′ ( x ) dx, (2.10) N ALZER’S INEQUALITY 5 holds, where T rap ( f ) := ( b − a ) f ( a ) + f ( b )2 − Z ba f ( x ) dx. The inequality is sharp.Proof.
Repeating the steps in the proof of Theorem 5 taking in account that norestrictions on f , we have0 ≤ Z ba " f ′ ( x ) f ( x ) − b − a ) · (cid:18) x − a + b (cid:19) dx = Z ba f ′ ( x ) f ( x ) dx − b − a ) f ( x ) · T rap ( f ) + 12( b − a )which gives that2 T rap ( f ) − ( b − a ) f ( x ) b − a · max a ≤ x ≤ b f ( x ) ≤ b − a · Z ba f ′ ( x ) dx, and thus the proof of (2.10) is established. The sharpness follows with f ( x ) =6 x − x + 1, for all x ∈ [0 , (cid:3) Another generalization for (2 n )-times differentiable functions is considered asfollows: Theorem 7.
Let f : I ⊂ R → R be (2 n ) -times differentiable ( n ≥ on I ◦ , theinterior of the interval I , where a, b ∈ I ◦ with a < b , such that f (2 n ) ∈ L [ a, b ] . If R ba f ( t ) dt = 0 , then the inequality k f k ∞ ≤ (cid:18) b − a (cid:19) n · (cid:13)(cid:13)(cid:13) f (2 n ) (cid:13)(cid:13)(cid:13) (2.11) holds, where, k f k ∞ := sup a ≤ x ≤ b | f ( x ) | and (cid:13)(cid:13) f (2 n ) (cid:13)(cid:13) = R ba (cid:12)(cid:12) f (2 n ) ( x ) (cid:12)(cid:12) dx .Proof. Setting α = (12) n ( b − a ) − ( n + )B (2 n + 1 , n + 1) n ∈ N , where B ( · , · ) is Euler-beta function. Assume that f attains its maximum value at x ∈ [ a, b ] and let sup a ≤ x ≤ b f ( x ) = f ( x ), for some a ≤ x ≤ b , then0 ≤ Z ba " f (2 n ) ( x ) f ( x ) − α · ( x − a ) n ( b − x ) n ( b − a ) n dx = Z ba (cid:0) f (2 n ) ( x ) (cid:1) f ( x ) dx − α ( b − a ) n f ( x ) Z ba ( x − a ) n ( b − x ) n f (2 n ) ( x ) dx + α ( b − a ) n Z ba ( x − a ) n ( b − x ) n dx M.W. ALOMARI
Therefore, Z ba (cid:16) f (2 n ) ( x ) (cid:17) dx ≥ α ( b − a ) n f ( x ) Z ba ( x − a ) n ( b − x ) n f (2 n ) ( x ) dx − α ( b − a ) n f ( x ) Z ba ( x − a ) n ( b − x ) n dx = 2 α ( b − a ) n f ( x ) Z ba ( x − a ) n ( b − x ) n f (2 n ) ( x ) dx (2.12) − α ( b − a ) f ( x ) B (2 n + 1 , n + 1) . It is not difficult to observe that Z ba ( x − a ) n ( b − x ) n f (2 n ) ( x ) dx = 0 , which follows by integrating by parts and using the given assumptions.Now, by triangle inequality we have Z ba (cid:12)(cid:12)(cid:12) f (2 n ) ( x ) (cid:12)(cid:12)(cid:12) dx ≥ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z ba (cid:16) f (2 n ) ( x ) (cid:17) dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≥ α ( b − a ) | f ( x ) | B (2 n + 1 , n + 1) , simple computations gives the required result (2.11). (cid:3) Useful Applications
Let f : I ⊆ R → R , be a twice differentiable mapping such that f ′′ ( x ) exists on I ◦ , and k f ′′ k ∞ = sup x ∈ ( a,b ) | f ′′ ( x ) | < ∞ . Then the trapezoid inequality (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( b − a ) f ( a ) + f ( b )2 − Z ba f ( x ) dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ( b − a ) k f ′′ k ∞ , (3.1)holds. Therefore, the integral R ba f ( x ) dx can be approximated in terms of thetrapezoidal rules, respectively such as: Z ba f ( x ) dx ∼ = ( b − a ) f ( a ) + f ( b )2 . By means of (2.10), it is significant to remark that the inequality has a trapezoidbound term, therefore we may rewrite (2.10) to obtain a new upper bound for thetrapezoid inequality, such as:
Corollary 2.
Under the assumptions of Theorem 6, we have T rap ( f ) ≤ b − a · max a ≤ x ≤ b f ( x ) + ( b − a ) · max a ≤ x ≤ b f ( x ) · Z ba f ′ ( x ) dx, (3.2) provided that max a ≤ x ≤ b f ( x ) = 0 . Equivalently, in terms of norms we may write |T rap ( f ) | ≤ b − a · k f k ∞ + ( b − a ) · k f ′ k k f k ∞ , (3.3) N ALZER’S INEQUALITY 7 where; k f k ∞ = sup a ≤ x ≤ b | f ( x ) | and k f ′ k = R ba | f ′ ( x ) | dx . The two inequalities aresharp. Henceforth, by setting M := 6( b − a ) · k f k ∞ + 12 ( b − a ) · k f ′ k k f k ∞ , a beautiful trapezoid inequality may be written as: (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( b − a ) f ( a ) + f ( b )2 − Z ba f ( x ) dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ( b − a ) M, (3.4)which holds with more less restrictions on f , and so if f is twice differentiableand has bounded second derivative, with M ≤ k f ′′ k ∞ , then totally (3.4) by itsassumptions can be better than (3.1), and exactly if M := k f ′′ k ∞ . So that wehave applied our result (2.10) to obtain new trapezoid type inequality which hasimportant applications in numerical integrations.One more direct interesting application is to bound the geometric mean by asharp upper bound. This happens if one assumes f ( a ) f ( b ) >
0, which alreadyholds by assumptions of Theorem 7, then (2.11) can be written as: p f ( a ) f ( b ) ≤ r b − a · Z ba f ′ ( x ) dx ! / , equivalently we write, G ( f ( a ) , f ( b )) ≤ r b − a · k f ′ k , (3.5)where G ( · , · ) is the geometric mean and the inequality is sharp.Moreover, if f is log-convex, i,e., f satisfies the inequality f ( λx + (1 − λ ) y ) ≤ f λ ( x ) f − λ ( y ) . for all x, y ∈ [ a, b ] and λ ∈ [0 , λ = , then the doubleinequality f ( A ( x, y )) ≤ G ( f ( x ) , f ( y )) ≤ r y − x · k f ′ k , (3.6)holds and sharp; provided that a ≤ x < y ≤ b , where A ( · , · ) is the arithmetic mean.Clearly, the left-hand side inequality sharp by the definition of log-convexity.A generalization of this result can be done if f is considered to be bijective on[ a, b ]. Choosing α, β ∈ [ a, b ] such that f ( α ) = f λ ( x ) and f ( β ) = f − λ ( y ), forsome λ ∈ [0 ,
1] and x, y ∈ [ a, b ]. Making use of (2.11) we have f ( α ) f ( β ) ≤ β − α Z βα f ′ ( x ) dx. (3.7)Therefore, a generalization of (3.6) may given as: f ( λx + (1 − λ ) y ) ≤ f λ ( x ) f − λ ( y ) ≤ f − (cid:0) f − λ ( y ) (cid:1) − f − (cid:0) f λ ( x ) (cid:1) · Z f − λ ( y ) f λ ( x ) f ′ ( x ) dx. (3.8) M.W. ALOMARI or written in terms of generalized means, as f ( A λ ( x, y )) ≤ G λ ( f ( x ) , f ( y )) ≤ f − (cid:0) f − λ ( y ) (cid:1) − f − (cid:0) f λ ( x ) (cid:1) · Z f − λ ( y ) f λ ( x ) f ′ ( x ) dx where, A λ ( x, y ) = λx +(1 − λ ) y , is the generalized arithmetic mean and G λ ( x, y ) = x λ y − λ , is the generalized geometric mean. References [1] H. Alzer, A continuous and a discrete varaint of Wirtinger’s inequality,
Mathematica Pannon-ica , 3 (1) 1992, 83–89.[2] P.R. Beesack, Integral inequalities involving a function and its derivative,
Amer. Math.Monthly , 78 (1971), 705–741.[3] P.R. Beesack, Extensions of Wirtinger’s inequality,
Trans. Royal Soc. Canada , 53 (1959),21–30.[4] W. Blaschke, Kreis und Kugel, Leipzig, 1916.[5] J.B. Diaz and F.T. Metcalf , Variations of Wirtinger’s inequality, in Inequalities (Edited byOved Shisha), Academic Press 79–103, 1967.[6] G. V. Milovanovi´c and I. ˇZ. Milovanovi´c, On a generaliztion of certain results of A. Ostrowskiand A. Lupa¸s, Univ. Beograd. Publ. Elektrotehn.
Fak. Ser. Mat. Fiz. , N. 634 - N. 677 (1979),62–69.
Department of Mathematics, Faculty of Science and Information Technology, IrbidNational University, P.O. Box 2600, Irbid, P.C. 21110, Jordan.
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