Weighted integral and integro-differential inequalities

Advances in Applied Mathematics 41 (2008) 227–246 www.elsevier.com/locate/yaama

Weighted integral and integro-differential inequalities Arcadii Z. Grinshpan Department of Mathematics and Statistics, University of South Florida, Tampa, FL 33620, USA Received 3 February 2007; accepted 14 September 2007 Available online 21 December 2007

Abstract A general weighted integral inequality for two continuous functions on an interval [a, b] is presented. The equality conditions are given. This result implies the new inequalities for the incomplete beta and gamma functions as well as the related estimates for the confluent hypergeometric function, error function, and Dawson’s integral. Also it implies various weighted integro-differential inequalities, those of the Opial type included, and some inequalities which involve the Erdélyi–Kober and Riemann–Liouville fractional integrals. © 2007 Elsevier Inc. All rights reserved. MSC: 26D10; 34A40; 26D15; 30A10; 33B20; 33B15; 33C15; 26A33 Keywords: Weighted integral inequalities; Integro-differential inequalities; Opial-type inequalities; Beta and gamma functions; Incomplete beta and gamma functions; Error function; Dawson’s integral; Confluent hypergeometric function; Erdélyi–Kober integral; Riemann–Liouville integral

1. Introduction We present a general weighted integral inequality for two continuous functions on an interval [a, b]. This inequality and its equality conditions are spotted due to the earlier works [32,36]. Using our general result we obtain the new inequalities for the incomplete beta and gamma functions and the incomplete confluent hypergeometric function. Also we show that this result implies various integro-differential inequalities, some sharp Opial-type inequalities in a general form included. In addition, the related estimates which involve the Erdélyi–Kober and Riemann–Liouville fractional integrals, the error function, the confluent hypergeometric function, and Dawson’s integral are given. It is well known that the above-mentioned “incomplete” E-mail address: [email protected]. 0196-8858/$ – see front matter © 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.aam.2007.09.002

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functions and integro-differential inequalities play an important role in probability theory, mathematical statistics, physics, financial mathematics, differential equations theory, and many other disciplines. The following theorem provides the basis to our approach. Theorem A. (See [32,36].) Let φ(t) and ψ(t) be complex-valued continuous functions on [0, 1]. For α, β, λ > 0, let functions L and R be defined by the formulas L(α, β, λ) =

(α + β) (α)(β)(λ)(α + β + λ)  1 2 1     α+β−1 λ−1  α−1 β−1 × τ (1 − τ )  t (1 − t) φ(τ t)ψ τ (1 − t) dt  dτ   0

(1)

0

and R(α, β, λ) =

1 (α + λ)(β + λ) 1 ×

t

α−1



β+λ−1 

(1 − t)

2 φ(t) dt ·

0

1

2  t β−1 (1 − t)α+λ−1 ψ(t) dt,

(2)

0

where  stands for Euler’s gamma function. Then the differences D(α, β, λ) defined by (2) and (1) satisfy the following inequalities: D(α, β, λ) = R(α, β, λ) − L(α, β, λ)  0

(3)

and D(α, β, λ)  λ(α + β + λ)D(α, β, λ + 1) + (α + λ)D(α + 1, β, λ) + (β + λ)D(α, β + 1, λ) λ(α + β) (α + 1)(β + 1)(λ)(α + β + λ + 1)  1 2 1     α α+β λ−1  β  × τ (1 − τ )  φ(τ t)ψ τ (1 − t) d t (1 − t)  dτ.  

+

0

(4)

0

The equality in (3), provided that φ and ψ are not identically 0, holds if and only if φ(t) = φ(0)eiθt and ψ(t) = ψ(0)eiθt for t ∈ [0, 1] and some real θ . Remarks. 1. Inequalities (3) and (4) hold for any piecewise continuous (finite-valued) functions φ(t) and ψ(t) on [0, 1]. Indeed, such functions can be approximated by the uniformly bounded

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continuous functions on this interval [32]. From (3) and (4) it follows that the equality in (3) for the piecewise continuous functions φ and ψ is possible only if 1

    φ(τ t)ψ τ (1 − t) d t α+m (1 − t)β+n = 0

(5)

0

for τ ∈ [0, 1] and m, n = 0, 1, . . . . 2. Note that λ(1 − τ )λ−1 and λ(λ) are equal to the delta function δ(1 − τ ) and 1, respectively, for λ = 0. Hence the limit case of (3) corresponds to the Cauchy–Bunyakovskii–Schwarz inequality as λ → 0. The inequality (3) as a non-trivial generalization of this classical result appears to be effective for a number of problems. Inequality (3) is proved in [32]; inequality (4) and Eq. (5) are used in [36] (see Theorem C and inequality (29) there) to obtain the equality conditions in (3); also see [31] for the discrete versions of (3) and (4). Some applications of Theorem A and its discrete predecessor are given in [31–36]. In particular, the structure of inequality (3) is convenient for us to deal with Euler’s beta and gamma functions as well as the generalized hypergeometric series and the related special functions and orthogonal polynomials. However a non-trivial extension of (3) is needed when we handle some other important cases. For example, it concerns certain estimates which involve the incomplete beta and gamma functions. Also inequality (3) results in the new weighted norm inequalities of different kind, various integral and differential inequalities, and inequalities for the bi-Hermitian forms (see [32–35]). So it is no surprise that its extension over an interval [a, b] leads to the Opial-type inequalities as they are related to the norm inequalities the same way the quadratic forms are related to the bilinear forms (see, e.g., Sinnamon [61]). As an example we give a generalization of the recent Troy’s theorem [64]. 2. Generalized weighted integral inequality Theorem B below gives a generalization of inequality (3) together with its equality conditions. The case of inequality (6) in this theorem when a = 0 and b = c = 1 corresponds to (3). Theorem B allows us to obtain a wide range of the weighted integral and integro-differential inequalities on an interval [a, b], not to say about a number of the estimates for the classical special functions. In this paper we restrict ourselves to the several examples and applications which are mentioned in the introduction and given in Section 3. Theorem B. Let f (t) and g(t) be complex-valued continuous functions on [a, b], where 0  a < b  c < ∞. Then for any numbers α, β, λ > 0, the following inequality holds: c τ

1−α−β

(c − τ )

a−b+c

 c1−α−β−λ ·

 τ +b−c 2     α−1 β−1 t (τ − t) f (t)g(c − τ + t) dt  dτ   

λ−1 

a

(α)(β)(λ)(α + β + λ) (α + λ)(β + λ)(α + β)

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b ×

t

α−1

2  f (t) dt ·

β+λ−1 

(c − t)

a

b

2  t α+λ−1 (c − t)β−1 g(t) dt.

(6)

a

The equality in (6), provided that f and g are not identically 0, holds if and only if a = 0, b = c, and f (t) = f (0)eiθt and g(t) = g(0)e−iθt (t ∈ [0, b], θ is a real number). Proof. Taking into account Remark 1 after Theorem A we use inequality (3) with   for t ∈ [a/c, b/c] and ψ(t) = g c(1 − t)

φ(t) = f (ct)

for t ∈ [1 − b/c, 1 − a/c];

also if a = 0 then φ(t) = 0 for t ∈ [0, a/c)

and ψ(t) = 0 for t ∈ (1 − a/c, 1],

and if b = c then φ(t) = 0

for t ∈ (b/c, 1] and ψ(t) = 0 for t ∈ [0, 1 − b/c).

We have 1 τ 1−α−β (1 − τ )λ−1 1−(b−a)/c

 τ +b/c−1 2        × t α−1 (τ − t)β−1 f (ct)g c(1 − τ + t) dt  dτ   a/c



(α)(β)(λ)(α + β + λ) (α + λ)(β + λ)(α + β)

b/c b/c 2 2   α−1 β+λ−1   f (ct) dt · t α+λ−1 (1 − t)β−1 g(ct) dt. (1 − t) × t a/c

(7)

a/c

Inequality (7) implies (6). If a = 0 and b = c, then the equality statement in Theorem B follows from that of in Theorem A. Now we show that inequality (6) is strict for any non-negative functions f and g, provided that they are not identically 0 and 0 < a < b  c or 0  a < b < c. Indeed, if such functions give the equality in (6) and a = 0 then Eq. (5) with m = 0 implies that τ +b−c 

  h(t, τ )d t α (τ − t)β+n = 0 (n = 0, 1, . . .),

(8)

a

where τ ∈ [a − b + c, c] and h(t, τ ) = f (t)g(c − τ + t)  0. Since the right-hand side of (6) is not 0, the equality in inequality (6) is possible only if its left-hand side is not 0 either. Hence there

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exists τ0 ∈ (a − b + c, c] such that h(t, τ0 ) is not identically 0 as a function of t ∈ [a, τ0 + b − c]. We define the moment sequence μn (n = 0, 1, . . .) by the formula τ0 +b−c

μn =

h(t, τ0 )t α−1 (τ0 − t)β+n−1 dt.

(9)

a

From (8) and (9) it follows that μn+1  (τ0 − a)μn and τ0 (β + n)μn = (α + β + n)μn+1 . We come to a contradiction as n → ∞. The case b = c is similar: if the equality in (6) holds, then Eq. (5) with n = 0 implies that τ +b−c 

  h(t, τ ) d t α+m (τ − t)β = 0 (m = 0, 1, . . .)

(10)

a

for τ ∈ [a − b + c, c]. Let the moment sequence νm (m = 0, 1, . . .) be defined by the formula τ0 +b−c

νm =

h(t, τ0 )t α+m−1 (τ0 − t)β−1 dt.

(11)

a

From (10) and (11) it follows that νm+1  (τ0 + b − c)νm and τ0 (α + m)νm = (α + β + m)νm+1 . Now we come to a contradiction as m → ∞.

2

Remark 3. As λ → 0 the limit of inequality (6) multiplied by λ gives the Cauchy–Bunyakovskii– Schwarz inequality for functions t (α−1)/2 (c − t)(β−1)/2 f (t)

and t (α−1)/2 (c − t)(β−1)/2 g(t).

The equality in this limit case holds if and only if functions f and g are proportional on [a, b].

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3. Examples and applications 3.1. Incomplete gamma function Theorem B and some properties of the coefficient convolutions which involve formal power series allow us to obtain a new parametrized inequality for the incomplete gamma functions (see, e.g., [1, Chapter 6], [24, p. 266]). The earlier results and related references can be found in Gautschi’s survey article [28] and, e.g., [5,23,25,27,29,30,40,42,44,53–55,62]. We use the following standard notation: b (x; a, b) =

t

x−1 −t

e

∞ (x; a) =

dt,

a

t x−1 e−t dt,

a

b γ (x; b) = (x) − (x; b) =

t x−1 e−t dt.

0

Also we use the generalized hypergeometric series 2 F0 [24, Chapters 4, 5], 2 F0 (α, β; -; z) =

∞  zn (α)n (β)n · , n! n=0

and the α-convolution u∗α of some power series u(z) =

∞

n=0 an z

n

[35]:

∞  an n z . u∗α (z) = (α)n n=0

To prove Corollary 1 we need an auxiliary result on the convolution of the product of two power series. As usual B(α, β) stands for Euler’s beta function. Lemma 1. (See [34,35].) Let u∗α (z) and v∗β (z) be analytic in a disk Dr = {z: |z| < r}, where u and v are some power series and α, β > 0. Then the (α + β)-convolution (uv)∗(α+β) (z) is analytic in Dr and the integral formula 1 B(α, β)(uv)∗(α+β) (z) =

  t α−1 (1 − t)β−1 u∗α (zt)v∗β z(1 − t) dt

0

holds for any z ∈ Dr . Corollary 1. For any x  0 and α, β, λ > 0, the following inequality holds:

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1 x

α+β

233

 2 t α+β−1 (1 − t)λ−1 e−xt H (t) dt

0



B(α + β + λ, λ) · γ (α; x)γ (β; x), B(α, β)B(α + λ, β + λ)

(12)

where the function H (t) =



2F 0

    α, (β + λ − 1)/2; -; t · 2 F0 β, (α + λ − 1)/2; -; t ∗(α+β)

(13)

is analytic for |t| < 1. Proof. We use inequality (6), where b = x, a > 0, c > x, f (t) = (c − t)(1−β−λ)/2 e−t/2

and g(t) = t (1−α−λ)/2 e(t−c)/2 .

We have c

τ 1−α−β (c − τ )λ−1 e−τ

a−x+c

 τ +x−c 2     α−1 β−1 (1−β−λ)/2 (1−α−λ)/2  × t (τ − t) (c − t) (c − τ + t) dt  dτ   a

 c1−α−β−λ ·

(α)(β)(λ)(α + β + λ) · (α; a, x)(β; c − x, c − a). (α + λ)(β + λ)(α + β)

(14)

Let a → 0+ and c → x+ in (14). It follows that 1 x

α+β

τ α+β−1 (1 − τ )λ−1 e−xτ

0

 1 2   (1−α−λ)/2   α−1 β−1 (1−β−λ)/2 1 − τ (1 − t) × t (1 − t) (1 − τ t) dt  dτ   0



B(α, β)B(α + β + λ, λ) · γ (α; x)γ (β; x). B(α + λ, β + λ)

We note that (1 − z)(1−β−λ)/2 =



  α, (β + λ − 1)/2; -; z ∗α



  β, (α + λ − 1)/2; -; z ∗β .

2F 0

and (1 − z)(1−α−λ)/2 =

2F 0

(15)

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Since both binomial functions above are analytic for |z| < 1, Lemma 1, (15) and (13) allow us to complete the proof. 2 To show some straightforward consequences of Corollary 1 we use the confluent hypergeometric function 1 F 1 (see, e.g., [1, Chapter 13], [24, Chapter 6]), 1 F 1 (α; β; z) =

∞  (α)n zn · , (β)n n! n=0

and the confluent hypergeometric limit function 0 F1 , 0 F1 (-; β; z) = lim 1 F 1 (α; β; z/α) = α→∞

∞  n=0

zn . (β)n n!

For α, β, λ > 0, let U (x, α, β, λ) =

x α+β B(α, β + λ)B(β, α + λ) γ (α; x)γ (β; x)

1 F 1 (α

+ β; α + β + λ; −x)

 (2αβ + (α + β)(λ − 1)) + 1 F 1 (α + β + 1; α + β + λ + 1; −x) (α + β + λ)

(16)

if x > 0,

and αβB(α, β + λ)B(β, α + λ)[2αβ + (α + β + 1)λ] U (0, α, β, λ) = (α + β + λ) (in particular, U (0, 1 − λ, 1 − λ, λ) = 1). We note that the function H (t) defined by (13) satisfies the inequalities      H (t)  0 F 1 -; α + β; αβ + (α + β)(λ − 1)/2 t  1 + αβ/(α + β) + (λ − 1)/2 t for t ∈ [0, 1), provided that α, β > 0 and α, β  1 − λ.

(17)

Hence it is easy to see that for any x  0, inequality (12) and (16) imply the inequalities x α+β B(α, β)B(α + λ, β + λ) U (x, α, β, λ)  B(α + β + λ, λ)γ (α; x)γ (β; x)

1

t α+β−1 (1 − t)λ−1 e−xt

0

   2 × 0 F 1 -; α + β; αβ + (α + β)(λ − 1)/2 t dt  1, 

(18)

provided that α, β, λ > 0, and condition (17) is satisfied. For example, if λ = 1 in (18), we have that 

(α + β) 2αβ · γ (α + β + 1; x) < · γ (α; x)γ (β; x) (19) B 2 (α, β) γ (α + β; x) + (α + β)x αβ

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for any x, α, β > 0. In its turn inequality (19) implies that αβB(α, β)  α + β

(α, β  0)

(20)

and (α)(β)  (α + β − 1) (α, β  1).

(21)

We mention that inequalities (20) and (21) are the consequences of monotonicity of the function F (x) = (x)/ (x + c) (c > 0) on [0, ∞) which is implied by the integral representation of the Euler psi function [24, p. 16], [20]. The related results and generalizations can be found in [3,4,6,7,22,37]. The case α = β = 1/2 in (18) is of interest because it gives a parametrized inequality which involves the error function erf(x) (see, e.g., [1, Chapter 7], [24, Chapter 6]), 2 erf(x) = √ π

x

e−t dt. 2

0

For any x  0 and λ  1/2, we obtain that x 1−λ e

−x 2

 1/2 x 2λ (λ − 1/2)t 1/λ (λ + 1/2) t 1/λ 1+ dt  erf(x). e · 2 (λ + 1) x

(22)

0

In particular, we have the following inequality

2

xD(x) π

1/2  erf(x)

(x  0),

where D(x) stands for Dawson’s integral [1, #7.1], D(x) = e

−x 2

x

2

et dt. 0

In addition, for any x, y > 0 and x1 ∈ [0, x), y1 ∈ [0, y), Theorem B with α = β = λ = 1/2, a = x1 , b = x, c = (xy − x1 y1 )/(y − y1 ), f (t) = e−t/2 and g(t) = exp{(y − y1 )t/[2(x − x1 )]} allows us to estimate the product (1/2; x1 , x)(1/2; y1 , y) from below (the case x1 = y1 = 0 results in an estimate for erf(x) erf(y)). The inequalities, which involvethe error function itself, α x various combinations of its values, and the related integrals of the type 0 e±t dt (α > 0), have been studied by many authors, see, e.g., [5,8,23,27,47,54,55] and the references therein. Since γ (x; b) can be expressed in terms of the confluent hypergeometric function [24, Chapter 6]: γ (x; b) = x −1 bx e−b 1 F 1 (1; x + 1; b) = x −1 bx 1 F 1 (x; x + 1; −b),

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several consequences of inequality (3) given in [32–34] for the function 1 F 1 lead to the corresponding inequalities for γ (x; b). In particular, this is true for the inequality + β; α + β + λ; −x) 1 1 F 1 (α; α + β + λ; −x)1 F 1 (β; α + β + λ; −x) 1 F 1 (α

V (x, α, β, λ) =

(23)

proved in [32] for any real x and α, β, λ  0 (α + β + λ > 0); see also [33,36]. The tightness of inequalities such as (18), (19), (22), (23) is of particular interest for the large values of x. We note that both functions U (x) and V (x), defined by (16) and (23), respectively, can be presented in the form    Δ 1 + O x −1 as x → ∞, where Δ = Δ(α, β, λ) = B(α + λ, β + λ)/B(α + β + λ, λ) < 1. See [24, Chapter 6] and [32] for the related asymptotic formulas and [20,37] for monotonicity of Δ as a function of λ (0 < λ < ∞). Thus the inequality |U | < 1 holds for all sufficiently large values of x even if condition (17) is not satisfied (this condition is certainly important for small x). The above asymptotics implies, in particular, that  1/2  1/2 = Δ(1/2, 1/2, λ) = lim U (x, 1/2, 1/2, λ)

x→∞

(λ + 1/2) [(λ)(λ + 1)]1/2

and 1/2  x 2λ x 1−λ −x 2 t 1/λ e dt = λ1/2 lim e x→∞ erf(x)

(λ > 0).

0

In other words, as x → ∞ the ratio of the left- and right-hand sides of (22) approaches (λ + 1/2) < 1 for any λ > 0. λ1/2 (λ) Note that the limit of this expression equals 1 as λ → ∞. 3.2. Incomplete beta function With f and g being the arbitrary linear combinations of some “convenient” functions (power, exponential, etc.), Theorem B generates various positive bi-Hermitian forms. For example, if f and g are some Müntz polynomials: f (t) =

n  k=1

ak t αk

and g(t) =

n  k=1

bk (1 − t)βk

(αk , βk  0; ak , bk ∈ C),

(24)

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then inequality (6) with a = 1 − x and b = c = 1 leads to a positive bi-Hermitian form in terms of the incomplete beta function (see, e.g., [1, Chapter 6], [24, p. 87]): x B(α, β; x) =

t α−1 (1 − t)β−1 dt

(α, β > 0; 0  x  1),

0

where B(α, β; 1) is Euler’s (complete) beta function B(α, β). To prove a more general result we use the weaker conditions αk > −α/2 and βk > −β/2 in (24) and take c > 1 and a > 0 in (6). Then we let c → 1+ and a → 0+ (or x → 1−). The corresponding result is presented in Corollary 2. Its particular case with x = 1 and αk = βk = k − 1 (k  n) was obtained in [32] in terms of the shifted factorials. Corollary 2. For any numbers α, β, λ > 0 and αk > −α/2, βk > −β/2 (k = 1, . . . , n; n = 1, 2, . . .), the following inequality holds: x t λ−1 (1 − t)α+β−1 0

 n  2  x − t   × ak bm (1 − t)αk +βm B β + βm , α + αk ;  dt  1−t  k,m=1



n (α)(β)(λ)(α + β + λ)  ak a m B(β + λ, α + αk + αm ; x) (α + λ)(β + λ)(α + β) k,m=1

×

n 

bk bm B(β + βk + βm , α + λ; x),

(25)

k,m=1

where x ∈ [0, 1] and ak , bk ∈ C. The equality in (25), provided that x > 0, αk , βk  0, and the Müntz polynomials f and g defined by (24) are not identically zero, holds if and only if both f and g are the constant functions and x = 1. The multiparameter inequality (25) for the incomplete beta function and its particular case in terms of Euler’s beta function, namely: 1 t

α+β−1

 n 2   αk +βm  ak bm B(α + αk , β + βm )t   dt  

λ−1 

(1 − t)

k,m=1

0



B(α, β)B(α + β + λ, λ) B(α + λ, β + λ) ×

n  k,m=1

ak a m B(α + αk + αm , β + λ) ·

n  k,m=1

bk bm B(β + βk + βm , α + λ),

(26)

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lead to a variety of estimates for these important functions. For example, the simplest case of (26), i.e. n = 1, shows that the function F (x, y) =

B 2 (α + x, β + y)B(2x + 2y, λ) B(2x, 2β + λ)B(2y, 2α + λ)

(x, y > 0),

where α, β, λ > 0, attains its maximum at (α, β). The case n = 2 is not obvious even if α1 = β1 = 0 and α2 = β2 = 1 in (26) [32]. The known estimates involving the beta function and the related references can be found, e.g., in [3,6,7,22]. Some inequalities for the incomplete beta function are given, e.g., in [45, Chapter 13], [39]. One can use the positive bi-Hermitian forms generated by inequality (6), for example, the one given by (25), in the study of the suitable sequence spaces. In fact, Theorem B, where f and g are the arbitrary Müntz polynomials divided by t α/2 and (1 − t)β/2 , respectively, and b < c, implies more general positive bi-Hermitian forms than (25). They can be expressed in terms of the differences b t α−1 (1 − t)β−1 dt

B(α, β; a, b) = B(α, β; b) − B(α, β; a) =

(0  a  b  1).

a

In the simplest case, when f and g are any non-zero constant functions on [a, b], we obtain Corollary 3. Corollary 3. For any α, β, λ > 0, the following inequality holds: 1 t α+β−1 (1 − t)λ−1 B 2 (α, β; x/t, 1 − y/t) dt x+y



(α)(β)(λ)(α + β + λ) B(α, β + λ; x, 1 − y)B(β, α + λ; y, 1 − x), (α + λ)(β + λ)(α + β)

(27)

where x, y  0 and x + y < 1. The equality in (27) holds if and only if x = y = 0. How “tight” are the above inequalities? We consider the simplest “incomplete” case. Let α = β = λ = 1. Then the difference of the right- and left-hand sides of (25) with n = 1 and α1 = β1 = 0 (or (27) with y = 0 and x replaced by 1 − x) equals ∞    x4 2 (1 − x) log(1 − x) + x + x 3 − x 2 /2 = x 3 + 3 k(k − 1)

(0 < x < 1).

k=4

If f and g in Theorem B are the linear combinations of the exponential functions we obtain certain integral inequalities which involve the incomplete confluent hypergeometric function: x M(, υ; z; x) =

t −1 (1 − t)υ−−1 ezt dt/B(, υ − ), 0

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where M(, υ; z; 1) = 1 F 1 (; υ; z) (0 <  < υ; 0  x  1). Also they involve the incomplete gamma function if α = β = 1 − λ in (6). It is worth mentioning that Beard [12–14] used the above incomplete integral and the incomplete gamma function for some actuarial applications. In the simplest case, when f (t) = e−yt/2 and g(t) = eyt/2 in (6) (y is real), we come to Corollary 4. Corollary 4. For any numbers α, β, λ > 0, the following inequality holds: 

x t

λ−1

(1 − t)

α+β−1 yt

e B

2

0

x−t β, α; 1−t

 dt

 B 2 (α, β)B(α + β, λ)M(β + λ, α + β + λ; y; x)M(β, α + β + λ; −y; x),

(28)

where y is any real number and x ∈ [0, 1]. The equality in (28) for x > 0 holds if and only if y = 0 and x = 1. Inequality (28) with x = 1 implies that 1 F 1 (λ; α

+ β + λ; y)  1 F 1 (β + λ; α + β + λ; y)1 F 1 (β; α + β + λ; −y)

(29)

for any real y. Kummer’s transformation [24, p. 253]: x 1 F 1 (α; β; x) = e 1 F 1 (β

− α; β; −x)

shows that inequality (29) is equivalent to (23). A combination of the exponential and power functions in (6), for instance, f (t) = e−yt and g(t) = (1 − t)η , also leads to a weighted integral inequality which involves the incomplete beta and incomplete confluent hypergeometric functions. The related consequences of inequality (3) can be found in [32,33]. 3.3. Integro-differential inequalities Inequality (6) in Theorem B is convenient for us to obtain some weighted estimates in terms of γ ,η the Erdélyi–Kober fractional integral operator Ia and the Riemann–Liouville fractional integral γ Ia of order γ (γ > 0): γ ,η Ia f (τ ) =

τ −γ −η (γ )

τ

(τ − t)γ −1 t η f (t) dt,

a γ Ia f (τ ) =

1 (γ )

τ

(τ − t)γ −1 f (t) dt

a

(see, e.g., [26,38,56,57]). These estimates lead to certain weighted Hardy inequalities and through them to various integro-differential inequalities. For example, let f (t) be replaced by f (t)t η−α+1 eyt/2 and g(t) = (c − t)γ −β e−yt/2 in (6). Then we obtain Corollary 5 which implies several inequalities of the above-mentioned type.

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Corollary 5. Let f (t) be a complex-valued continuous function on [a, b], where 0  a < b  c < ∞. Then for any numbers α, β, λ, γ > 0, and real η and y (a > 0 if α  2η + 2; b < c if β  2γ ), the following inequality holds: 2 b τ b  2    γ −1 η t f (t) dt  u(τ ) dτ  A f (t) v(t) dt,  (τ + c − t − b)   a

a

(30)

a

where u(t) = (t − b + c)1−α−β (b − t)λ−1 eyt ,

v(t) = t 2η−α+1 (c − t)β+λ−1 eyt ,

and A is a constant independent of f , A  K(α, β, λ, γ , y) = b ×

eyb cα+β+λ−1

·

(α)(β)(λ)(α + β + λ) (α + λ)(β + λ)(α + β)

t α+λ−1 (c − t)2γ −β−1 e−yt dt.

a

The equality in (30), provided that α  η + 1, β  γ , and f is not identically 0, holds if and only if α = η + 1, γ = β, a = y = 0, b = c, and f is a constant function. Inequality (30) with b = c involves the Erdélyi–Kober integral. If, in addition, η = 0, then we obtain a weighted estimate of the Riemann–Liouville integral. Inequality (30) with η = 0 and γ = 1 is a weighted Hardy inequality. The weighted Hardy inequalities have been characterized by Muckenhoupt [49,50] except for the best constants. The more general weighted inequalities, which involve the Riemann–Liouville integral and generalize the well-known Hardy–Littlewood theorem [38, Theorem 329], are given, e.g., in [11,46,63]. It is interesting to compare the expression for K in (30) and the least suitable constant A, which is available in the theory of weighted Hardy inequalities and their extensions. Clearly, the range of the basic parameters in (30) is important. For instance, if η = 0 and γ = 1, then according to [49] we come to what seems to be the best upper bound (in this theory) for A in (30):  b u(t) dt ·

A  4 sup a<τ
τ



τ v

−1

(t) dt .

(31)

a

Here are some examples with the simple weights u and v and parameters a = y = 0, c = b, and λ = 1. First, let α = β = 1/2 and b = 1. Then u(t) = 1, v(t) = [t (1−t)]1/2 , and K = π/2 in (30), but the right-hand side of (31) is greater than π . If α = β = 1, then K = 1 and inequality (30) with u(t) = t −1 and v(t) = (b − t) is sharp. From (31) we have that A  4(log 2)2 = 1.92 . . . . So the explicit formula for K in (30) is preferable in both cases. However if α = 1 and β → 2−, then K → ∞ though inequality (31) with u(t) = t −2 and v(t) = (b − t)2 gives a special case of the above-mentioned Hardy–Littlewood theorem: A  4b−2 . In addition, we note that a slight modification of the related result in [46] gives an extension of (31) for γ > 1, which can be used for further comparison of the constants.

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241

The above results and examples encourage us to apply Corollary 5 and Theorem B itself in the field of weighted integro-differential inequalities. Many known integro-differential inequalities and related references can be found, e.g., in [38,47,48]. We start with Corollary 6 which is implied by Corollary 5 with γ = 1, η = 0, and f (t) = h (t), where h is some continuously differentiable function. Corollary 6. Let h(t) be a complex-valued continuously differentiable function on [a, b], where 0 < a < b < c < ∞. Then for any numbers α, β, λ > 0 and real y, the following inequality holds: b

  h(t) − h(a)2 u(t) dt  K

a

b

  2 h (t) v(t) dt,

(32)

a

where K, u(t), and v(t) are defined in (30) with γ = 1 and η = 0. The equality in (32) holds if and only if h is a constant function. Also inequality (32) holds for a = 0 if α ∈ (0, 2) or if the function [h(t) − h(0)]t −α is continuously differentiable on [0, b], and it holds for c = b if β ∈ (0, 2). Then the equality in (32), provided that α, β ∈ (0, 1], and h is not a constant function, holds if and only if a = y = 0, c = b, β = 1, and h(t 1/α ) is a linear function of t. Corollary 7. Let h(t) be a complex-valued continuously differentiable function on [a, b] (0  a < b), such that h(a) = h(b) = 0. Then for any x ∈ (a, b), λ > 0, and real y, the following inequality holds: b a

2  πeyx (λ)(λ + 1) |t − x|λ−1 eyt h(t) dt  λ · 2 b  (λ + 1/2)  x ×

t

λ−1/2

1/2 −yt

(b − t)

e

x dt ·

a

a

b +

t

2  t 1/2 (b − t)λ−1/2 eyt h (t) dt

1/2

λ−1/2 −yt

(b − t)

e

b dt ·

x

t

λ−1/2



 2  h (t) dt .

1/2 yt  

(b − t)

e

(33)

x

In particular, if a = y = 0 and λ = 1, then the inequality b 0

  h(t)2 dt  πb 4

b

 1/2   2 h (t) dt t (b − t)

(34)

0

holds. The equality in (33) and (34) holds if and only if h is identically 0. Proof. We apply Corollary 6 to the function h(b − t), t ∈ [0, b − a]. Also we replace c and y in (32) by b and −y, respectively. It follows that

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b

  h(t)2 u1 (t) dt  K1

a

b

  2 h (t) v1 (t) dt,

where

a

u1 (t) = (a + b − t)

1−α−β

(t − a)λ−1 eyt ,

v1 (t) = t β+λ−1 (b − t)1−α eyt , and eya (α)(β)(λ)(α + β + λ) K1 = α+β+λ−1 · (α + λ)(β + λ)(α + β) b b ×

t 1−β (b − t)α+λ−1 e−yt dt.

(35)

a

Now we use inequalities (32) with c = b = x and (35) with a = x. We put α = β = 1/2 in both inequalities. It is clear that the equality in (35) with β = 1 holds if and only if h is identically 0. 2 It is well known that the starting point of many publications on the integro-differential inequalities of the Opial type (see, e.g., [2,9,10,15–19,21,41,43,47,48,58–61,64]) is the sharp inequality b

  h(t)h (t) dt  b 4

0

b

  2 h (t) dt

  h(0) = h(b) = 0

(36)

0

proved by Opial [52] under some additional conditions and reproved by Olech [51] for all real, absolutely continuous functions h(t) on [0, b]. Due to the important applications the sharpness of the Opial-type inequalities and their connection to the other integral inequalities are of interest. The elegant Beesack theorem implies inequality (36). Beesack’s theorem. (See [15].) Let h(t), h(0) = 0, be real and absolutely continuous on [0, b]. Then b

  h(t)h (t) dt  b 2

b

0

  2 h (t) dt.

(37)

0

Equality in (37) holds if and only if h(t) = At on [0, b], where A is a constant. Troy gave the following generalization of Beesack’s inequality in terms of the complexvalued, continuously differentiable functions and then he used it to generalize the results of Opial and Olech. Troy’s theorem. (See [64].) Let h(t) be a complex-valued, continuously differentiable function on [a, b], a  0, and let p > −1. Then b a

  1 t p h(t)h (t) dt  √ 2 p+1

b a

2  p+1  b − at p h (t) dt

(38)

A.Z. Grinshpan / Advances in Applied Mathematics 41 (2008) 227–246

243

if h(a) = 0, and b a

  1 t h(t)h (t) dt  √ 2 p+1 p

b

2  p  bt − a p+1 h (t) dt

(39)

a

if h(b) = 0. Troy announced the sharpness of (38) and (39) as an open problem [64, R2]. Bravyi and Gusarenko [16] analyzed the sharpness of his result in some cases and found the best constants for the extended inequalities with the boundary conditions of different type. The paper [64], in particular, inequality (38) was an inspiration for the work of Anastassiou, G. Goldstein, and J. Goldstein on multidimensional Opial inequalities [10]. Note that Corollaries 6 and 7 imply various Opial-type inequalities. For example, inequality (32) (and (35)) coupled with the arithmetic mean-geometric mean inequality (or Cauchy–Bunyakovskii–Schwarz inequality) implies a generalization of Beesack’s inequality (37). Though each of the inequalities (36)–(39) for b = 1 immediately implies the general case of the corresponding inequality we prefer the traditional style with an arbitrary b which opens the gate to the new results. We observe that Troy’s approach in [64] is entirely based on the Cauchy–Bunyakovskii– Schwarz and arithmetic mean-geometric mean inequalities. It seems reasonable to arrange his proofs of inequalities (38) and (39) in terms of the functions and parameters of Theorem B. Namely, f (t) = |t − d|1/2 |h (t)| for t ∈ [a, b], where d = a in the proof of inequality (38) and d = b in the proof of inequality (39). In the proofs of (38) and (39), the function g(t) on [a, b] equals |t − d|−1/2 |h(t)|, where d = a and b, respectively. As λ → 0 inequality (6) with α = p + 1, β = 1, and the chosen continuous functions f and g corresponds to the basic inequalities in [64] on the interval [a, b]. For λ > 0 and the above functions f and g, the equality conditions in Theorem B imply that inequality (6) is strict if the function h is not identically 0. Keeping all these factors, Remark 3, and the known inequality

  h(τ )2  (τ − d)

τ

  2 h (t) dt,

h(d) = 0,

(40)

d

in mind we obtain a generalization of Troy’s theorem as a consequence of Theorem B. Note that (40) as a consequence of the Cauchy–Bunyakovskii–Schwarz inequality is implied by inequality (32) as λ → 0. Corollary 8. Let h(t) be a complex-valued, continuously differentiable function on [a, b], 0  a < b  c < ∞, such that h(d) = 0, d ∈ [a, b]. Then for any numbers α > 0 and λ  0, the following inequality holds:

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c

λ(c − τ )λ−1 τ −α

αcα+λ a−b+c

 τ +b−c    α−1  × t  a

b

2 1/2     t −d  h (t)h(c − τ + t) dt dτ c−τ +t −d

 2 t α−1 (c − t)λ |t − d|h (t) dt

 a

 d ×

2  α+λ  t − a α+λ h (t) dt +

a

b

   α+λ  2 b − t α+λ h (t) dt ,

(41)

d

where λ(c − τ )λ−1 = δ(c − τ ) if λ = 0. The equality in (41), provided that h is not identically 0, holds if and only if λ = 0 and h(t) = A(t − d) on [a, b], where A is a constant. To compare inequality (41) with Troy’s inequalities (38) and (39) one can replace its righthand side with the greater expression 1 4

 b

 2 t α−1 (c − t)λ |t − d|h (t) dt

a

d + a

2  α+λ  t − a α+λ h (t) dt +

b

2  α+λ  b − t α+λ h (t) dt

2 .

(42)

d

From (41) and (42) it follows that the equality in (38) and (39), provided that h is not identically 0, holds if and only if p = 0 and h(t) is the corresponding linear function on [a, b] (see the open problem in [64, R2]). References [1] M. Abramowitz, I.A. Stegun (Eds.), Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover, New York, 1992, reprint of the 1972 edition. [2] R.P. Agarwal, P.Y. Pang, Opial Inequalities with Applications in Differential and Difference Equations, Kluwer, Dordrecht, 1995. [3] R.P. Agarwal, N. Elezovi´c, J. Peˇcari´c, On some inequalities for beta and gamma functions via some classical inequalities, J. Inequal. Appl. 5 (2005) 593–613. [4] H. Alzer, On some inequalities for the gamma and psi functions, Math. Comp. 66 (1997) 373–389. [5] H. Alzer, On some inequalities for the incomplete gamma function, Math. Comp. 66 (1997) 771–778. [6] H. Alzer, Sharp inequalities for the beta function, Indag. Math. (N.S.) 12 (1) (2001) 15–21. [7] H. Alzer, Some beta-function inequalities, Proc. Roy. Soc. Edinburgh Sect. A 133 (4) (2003) 731–745. [8] H. Alzer, Functional inequalities for the error function, Aequationes Math. 66 (1–2) (2003) 119–127. [9] G.A. Anastassiou, General fractional Opial type inequalities, Acta Appl. Math. 54 (1998) 303–317. [10] G.A. Anastassiou, G.R. Goldstein, J.A. Goldstein, Multidimensional Opial inequalities for functions vanishing at an interior point, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 15 (1) (2004) 5–15.

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