Bulletin T.CXXXIII de l’Académie serbe des sciences et des arts − 2006 Classe des Sciences mathématiques et naturelles Sciences mathématiques, N\underline{o} 31

INEQUALITIES WHICH INCLUDE q-INTEGRALS^{1}

M. S. STANKOVIĆ, P. M. RAJKOVIĆ, SLADJANA D. MARINKOVIĆ

(Presented at the 1st Meeting, held on February 24, 2006)

A b s t r a c t. The main problem in analyzing inequalities which include q-integrals is the fact that q-integral of a function over an interval [a, b] (0 < a < b) is defined by the difference of two infinite sums. Thus defined qintegral properties must include the points outside of interval of integration.

In this paper, we will signify to some directions for solving this problem and derive some inequalities which are analogues to well-known ones in standard integral calculus.

AMS Mathematics Subject Classification (2000): 33D60, 26D15 Key Words: integral inequalities, q-integral

1. Introduction

In the fundamental books about q-calculus [3],[4] the q-integral of the function f over the interval [0, b] is defined by

I_{q}(f; 0, b) = ∫_{0}^{b} f(x)d_{q}x = b(1 − q) ∑_{n=0}^{∞} f(bq^{n})q^{n} (0 < q < 1). (1)

1This paper was presented at the Conference GENERALIZED FUNCTIONS 2004, Topics in PDE, Harmonic Analysis and Mathematical Physics, Novi Sad, September 22– 28, 2004

This research was supported by the Science Foundation of Republic Serbia, Project No. 144023 and Project No. 144013

If f is integrable over [0, b], then

lim_{q↗1} I_{q}(f; 0, b) = ∫_{0}^{b} f(x) dx = I(f; 0, b).

Our attention was pulled up by the definition of integral over the interval [a, b]. Namely, generally accepted definition is

I_{q}(f; a, b) = ∫_{a}^{b} f(x)d_{q}x = ∫_{0}^{b} f(x)d_{q}x − ∫_{0}^{a} f(x)d_{q}x (0 < q < 1). (2)

For example, in that case the values of q-integrals of the polynomials over [a, b] are very similar to well-known ones in the standard integral calculus. But, problem is what will happen if f is defined in [a, b] and if it is not defined in [0, a].

In this paper we specify two ways to overcome the mentioned problem. The first one is the restriction of the q-integral over [a, b] to a finite sum whose number of the elements directly depends on a, b and q (see [2]). The second one is indicated in [6] and it means introduction of the definition of the q-integral of the Riemann type.

2. The q-integrals, correlations and properties

Let a, b and q be some real numbers such that 0 < a < b and q ∈ (0, 1). Beside the q-integrals defined by (1) and (2), we will consider two other

types of the q-integrals.

In the paper [2], H. Gauchman has introduced the restricted q-integral

G_{q}(f; a, b) = ∫_{a}^{b} f(x) d_{q}^{G}x = b(1 − q) ∑_{k=0}^{n−1} f(bq^{k})q^{k} (a = bq^{n}). (3)

Let us notice that lower bound of integral is a = bq^{n}, i.e., it is tied by chosen q,b and positive integer n.

In the paper [6], we have introduced Riemann-type q-integral by

R_{q}(f; a, b) = ∫_{a}^{b} f(t)d_{q}^{R}t = (b − a)(1 − q) ∑_{k=0}^{∞} f(a + (b − a)q^{k})q^{k}. (4)

This definition includes only points within the interval of the integration.

Inequalities which include q-integrals 139

The different types of the q-integral defined by (1)–(4) can be denoted in the unique way by J_{q}( ·; a_{(J)}, b), where J can be G, I or R. Interval of the integration E_{(J)} = [a_{(J)}, b] of q-integral J_{q}( ·; a_{(J)}, b) depends on its type:

a_{(G)} = bq^{n}, n ∈ ℕ, for G_{q}( ·; a, b);

a_{(I)} = 0 for I_{q}( ·; 0, b);

a_{(I)} and a_{(R)} are arbitrary numbers a ∈ [0, b] for I_{q}( ·; a, b) and R_{q}( ·; a, b). We can say that a real function f is q-integrable on [0, b] or [a, b] if the

series in (1) and (2) converge. In the similar way, we say that f is qRintegrable on [a, b] if the series in (4) converges. From now on, it will be assumed that the function f is q-integrable on [0, b] (qR-integrable on [a, b]) whenever I_{q}(f; 0, b) or I_{q}(f; a, b) (R_{q}(f; a, b)) appears in the formula.

In this research it is convenient to define the operators

: f ↦ f̂,  f̂(x) = f(a + (b − a)x),
: f ↦ f̃,  f̃(x) = bf(bx) − af(ax),
˘ : f ↦ f̆,  f̆(x) = f(bx) − f(ax),

such that associate the functions defined on [0, 1] to the function defined on [a, b]. Notice that, for x ∈ [0, 1], it is

ˆ(fg)(x) = f̂(x) ĝ(x), ˜(fg)(x) = \frac{1}{b − a}(f̃(x)g̃(x) − ab f̆(x)ğ(x)). (5)

The correlations between the q-integrals defined by (1)–(4) are given in the following lemma.

Lemma 2.1. If the real function f is q-integrable on [0, b] or qR-integrable on [a, b], 0 < a < b, then it holds

I_{q}(f; 0, b) = lim_{n→∞} G_{q}(f; bq^{n}, b),  (6)
I_{q}(f; a, b) = I_{q}(f̃; 0, 1),  (7)
R_{q}(f; a, b) = (b − a)I_{q}(f̂; 0, 1).  (8)

P r o o f. The relation (6) is evident because G_{q}(f; bq^{n}, b), n ∈ ℕ, are the partial sums of the series I_{q}(f; 0, b). The equality (7) is valid according to

I_{q}(f; a, b) = (1 − q) ∑_{k=0}^{∞}(bf(bq^{k}) − af(aq^{k}))q^{k} = I_{q}(f̃; 0, 1).

Finally, for (8), according to the definition we have

R_{q}(f; a, b) = (b − a)(1 − q) ∑_{k=0}^{∞} f(a + (b − a)q^{k})q^{k} = (b − a)I_{q}(f̂; 0, 1). □

The mentioned connections can be used to derive the inequalities for all types of the q-integrals. By (6), the inequalities for the infinite sum I_{q}(f; 0, b) can be derived in the limit process from this one for G_{q}(f; a, b), defined by the finite sum. Using (7) and (8), the integrals I_{q}(f; a, b) and R_{q}(f; a, b) can be considered as the q-integrals over [0, 1]. Nevertheless, the results for I_{q}(f; a, b) are quite rough because the points outside of the interval of the integration (i.e., points on [0, a]) are included.

According to (5) and Lemma 2.1, the following integral relations are valid:

R_{q}(fg; a, b) = (b − a)I_{q}( ˆ(fg); 0, 1) = (b − a)I_{q}(f̂ ĝ; 0, 1),  (9)
I_{q}(fg; a, b) = I_{q}( ˜(fg); 0, 1) = \frac{1}{b − a}(I_{q}(f̃ g̃; 0, 1) − ab I_{q}(f̆ ğ; 0, 1)). (10)

At last, let us remind on some definitions and terms from q-calculus. The q-natural number is defined by

[n]_{q} = \frac{1 − q^{n}}{1 − q} = 1 + q + ⋯ + q^{n−1}, n ∈ ℕ.

The function f : [a, b] → ℝ is called q-increasing (q-decreasing) on [a, b] if f(qx) ≤ f(x) (f(qx) ≥ f(x)) whenever x ∈ [a, b] and qx ∈ [a, b]. It is easy to see that if the function f is increasing (decreasing), then it is q-increasing (q-decreasing) for 0 < q < 1.

3. q-Chebyshev inequality

In this section we give the q-analogues of Chebyshev inequality for the monotonic functions (see [5], pp. 239.). The discrete case of this inequality is used in [2] for the restricted q-integrals. We derive its variants for the rest of the q-integrals.

Theorem 3.1. Let f, g : E_{(J)} → ℝ be two real functions, both q-decreasing or both q-increasing. If J_{q}( ·; a_{(J)}, b) is the q-integral defined by (1), (3) or (4), it holds

J_{q}(fg; a_{(J)}, b) ≥ \frac{1}{b − a_{(J)}} J_{q}(f; a_{(J)}, b) J_{q}(g; a_{(J)}, b).

P r o o f. For J_{q}( ·; a_{(J)}, b) = G_{q}( ·; a, b), a = bq^{n}, the inequality is proved in [2]. So, the inequalities

G_{q}(fg; bq^{n}, b) ≥ \frac{1}{b − bq^{n}} G_{q}(f; bq^{n}, b) G_{q}(g; bq^{n}, b)

are valid for all n = 1, 2, …. When n → ∞, using (6) we get the desired inequality for J_{q}( ·; a_{(J)}, b) = I_{q}( ·; 0, b). In the case J_{q}( ·; a_{(J)}, b) = R_{q}( ·; a, b), from the q-monotonicity of the functions f and g on [a, b] follows the qmonotonicity of the functions f̂ and ĝ on [0, 1]. Hence, we have

I_{q}(f̂ ĝ; 0, 1) ≥ I_{q}(f̂; 0, 1) I_{q}(ĝ; 0, 1).

According to (8) and (9) we get the required inequality. □

The Chebyshev inequality in the source form is not valid for I_{q}( ·; a, b), where 0 < a < b.

Example 3.1 For f(x) = x^{3} and g(x) = x^{4} on the interval [1, 2] we have

I_{q}(x^{3} · x^{4}; 1, 2) − I_{q}(x^{3}; 1, 2)I_{q}(x^{4}; 1, 2) = 255\frac{1 − q}{1 − q^{8}} − 465\frac{(1 − q)^{2}}{(1 − q^{4})(1 − q^{5})},

wherefrom we conclude that the inequality holds only for q > 1/2, but it has opposite sign for q < 1/2.

Lemma 3.2. Let the function f : [0, b] → ℝ be increasing and 0 < a < b. If there exist two positive constants l and L such that a^{2}/b^{2} ≤ l/L and for every x, y ∈ [0, b] the inequality

l ≤ \frac{f(x) − f(y)}{x − y} ≤ L

is valid, then the function f̃: [0, 1] → ℝ is increasing too.

P r o o f. Under the conditions of the Lemma, for every 0 ≤ x < y ≤ b we have

l(y − x) ≤ f(y) − f(x) ≤ L(y − x).

Then it holds

\begin{aligned}f̃(y) − f̃(x) & = b(f(by) − f(bx)) − a(f(ay) − f(ax)) \\ & ≥ (b^{2}l − a^{2}L)(y − x) ≥ 0. & □ \end{aligned}

Theorem 3.3. Let f, g : [0, b] → ℝ be two real increasing functions. If there exist the constants l_{f}, L_{f}, l_{g} and L_{g} such that a^{2}/b^{2} ≤ l_{f}/L_{f}, a^{2}/b^{2} ≤ l_{g}/L_{g} and

l_{f} ≤ \frac{f(x) − f(y)}{x − y} ≤ L_{f}, l_{g} ≤ \frac{g(x) − g(y)}{x − y} ≤ L_{g}

holds, then the inequalities are valid:

\begin{aligned}(a) & I_{q}(fg; a, b) & ≥ \frac{1}{b − a}I_{q}(f; a, b)I_{q}(g; a, b) − \frac{ab(b − a)}{[3]_{q}}L_{f}L_{g} \\ (b) & I_{q}(fg; a, b) & ≥ \frac{1}{b − a}I_{q}(f; a, b)I_{q}(g; a, b) − \frac{ab}{b − a}(f(b) − f(0))(g(b) − g(0)). \end{aligned}

P r o o f. Suppose that f and g are both increasing on [0, b]. Then, according to Lemma 3.2, f̃ and g̃ are both increasing and hence q-increasing on [0, 1]. With respect to (10) we can write

I_{q}(fg; a, b) = \frac{1}{b − a}(I_{q}(f̃ g̃; 0, 1) − ab I_{q}(f̆ ğ; 0, 1)).

Using Theorem 3.1, we have

I_{q}(f̃ g̃; 0, 1) ≥ I_{q}(f̃; 0, 1) I_{q}(g̃; 0, 1),

wherefrom

I_{q}(fg; a, b) ≥ \frac{1}{b − a}(I_{q}(f; a, b) I_{q}(g; a, b) − abI_{q}(f̆ ğ; 0, 1)).  (11)

(a) Under the conditions satisfied by the functions f and g on [0, b], it holds

\begin{aligned}I_{q}(f̆ ğ; 0, 1) & = (1 − q) ∑_{k=0}^{∞}(f(bq^{k}) − f(aq^{k}))(g(bq^{k}) − g(aq^{k}))q^{k} \\ & ≤ (1 − q) ∑_{k=0}^{∞} L_{f}L_{g}(bq^{k} − aq^{k})^{2}q^{k} = L_{f}L_{g}(b − a)^{2} \frac{1 − q}{1 − q^{3}} \end{aligned}

Substituting this estimation in (11), we get the first inequality.

(b) Since the functions f and g are increasing on [0, b], it holds

I_{q}(f̆ğ; 0, 1) ≤ (1−q)(f(b)−f(0))(g(b)−g(0)) ∑_{k=0}^{∞} q^{k} = (f(b)−f(0))(g(b)−g(0)),

what, with (11), gives the second inequality. □

4. q-Chebyshev functional and inequalities

Let us denote by

T (f, g; a, b) = \frac{1}{b − a} ∫_{a}^{b} f(x)g(x)dx − \frac{1}{(b − a)^{2}} ∫_{a}^{b} f(x)dx ∫_{a}^{b} g(x)dx

the Chebyshev functional, i.e., the functional whose positivity for the monotonic functions proves Chebyshev inequality. Its q-analogues can be made for each of mentioned types of q-integrals:

T_{q}^{(J)}(f, g; a_{(J)}, b) = \frac{J_{q}(fg; a_{(J)}, b)}{b − a_{(J)}} − \frac{J_{q}(f; a_{(J)}, b)J_{q}(g; a_{(J)}, b)}{(b − a_{(J)})^{2}},

where J_{q}( ·; a_{(J)}, b) is one of q-integrals defined by (1)–(4).

With respect to Lemma 2.1, it can be evince that

T_{q}^{(I)}(f, g; 0, b) = lim_{n→∞} T_{q}^{(G)}(f, g; bq^{n}, b),  (12)
T_{q}^{(R)}(f, g; a, b) = T_{q}^{(I)}(f̂, ĝ; 0, 1),  (13)
T_{q}^{(I)}(f, g; a, b) = \frac{1}{(b − a)^{2}} (T_{q}^{(I)}(f̃, g̃; 0, 1) − ab I_{q}(f̆ğ; 0, 1)). (14)

In that way, the basic inequalities including the Chebyshev functional are derived for T_{q}^{(G)}(f, g; a, b), and then for the other variants.

Theorem 4.1. Let f, g : E_{(J)} → ℝ be two real functions. If there exist the real constants m and M such that

m(g(x) − g(y)) ≤ f(x) − f(y) ≤ M(g(x) − g(y)), a_{(J)} ≤ x < y ≤ b,

is valid, then it holds the inequality

(m + M) T_{q}^{(J)}(f, g; a_{(J)}, b) ≥ T_{q}^{(J)}(f, f; a_{(J)}, b) + mM T_{q}^{(J)}(g, g; a_{(J)}, b),

where J_{q}( ·; a_{(J)}, b) is the q-integral defined by (1), (3) or (4).

P r o o f. At the start, let J_{q}( ·; a_{(J)}, b) = G_{q}( ·; a, b), a = bq^{n}, n ∈ ℕ. Under the conditions satisfied by the functions f and g, the product

(M(g(bq^{i})−g(bq^{j}))−(f(bq^{i})−f(bq^{j})))((f(bq^{i})−f(bq^{j}))−m(g(bq^{i})−g(bq^{j}))) is nonnegative for i, j = 0, 1, …, n − 1 (i < j), i.e.,

\begin{aligned}(f(bq^{i}) − f(bq^{j}))^{2} & +mM(g(bq^{i}) − g(bq^{j}))^{2} \\ & ≤ (m + M)(f(bq^{i}) − f(bq^{j}))(g(bq^{i}) − g(bq^{j})). \end{aligned}

Multiplying by q^{i+j} (i < j) and summing over i and j, we obtain

\begin{matrix}∑_{{i,j=0 \atop i<j}}^{n−1}(f(bq^{i}) − f(bq^{j}))^{2}q^{i+j} + mM ∑_{{i,j=0 \atop i<j}}^{n−1}(g(bq^{i}) − g(bq^{j}))^{2}q^{i+j} \\ ≤ (m + M) ∑_{{i,j=0 \atop i<j}}^{n−1}(f(bq^{i}) − f(bq^{j}))(g(bq^{i}) − g(bq^{j}))q^{i+j}. \end{matrix}

Since (see [1],[5]) T_{q}^{(G)}(f, g; a, b) can be presented in the form

T_{q}^{(G)}(f, g; a, b) = (\frac{b(1 − q)}{b − a})^{2} ∑_{{i,j=0 \atop i<j}}^{n−1}(f(bq^{i}) − f(bq^{j}))(g(bq^{i}) − g(bq^{j}))q^{i+j},

we have the required inequality.

For J_{q}( ·; a_{(J)}, b) = I_{q}( ·; 0, b) it is enough to put n → ∞ in the proved inequality for the previous case.

Finally, let J_{q}( ·; a_{(J)}, b) = R_{q}( ·; a, b). Under the condition satisfied by the functions f and g on [a, b], the functions f̂ and ĝ satisfy the same conditions on [0, 1]. Applying (13) and the proved inequality for T_{q}^{(I)}(f̂, ĝ; 0, 1), we get the statement. □

The upper inequality can be presented in the form

J_{q}((f −mg)(Mg−f); a_{(J)}, b) ≥ \frac{1}{b − a_{(J)}} J_{q}(f −mg; a_{(J)}, b)J_{q}(Mg−f; a_{(J)}, b),

what is Chebyshev inequality for the monotonic functions f−mg and Mg−f.

Theorem 4.2. Let f, g : E_{(J)} → ℝ be two real functions. If there exist the real constants l and L such that

\begin{aligned}l((b − a_{(J)})g(x) − J_{q}(g; a_{(J)}, b)) & ≤ (b − a_{(J)})f(x) − J_{q}(f; a_{(J)}, b) \\ & ≤ L((b − a_{(J)})g(x) − J_{q}(g; a_{(J)}, b)), \end{aligned}

on [a, b], then it holds the inequality

(l + L) T_{q}^{(J)}(f, g; a_{(J)}, b) ≥ T_{q}^{(J)}(f, f; a_{(J)}, b) + lL T_{q}^{(J)}(g, g; a_{(J)}, b),

where J_{q}( ·; a_{(J)}, b) is the q-integral defined by (1), (3) or (4).

P r o o f. Let J_{q}( ·; a_{(J)}, b) be G_{q}( ·; a, b), a = bq^{n}, or I_{q}( ·; 0, b). Then Chebyshev functional T_{q}^{(J)} can be represented (see [1]) in the form

\begin{matrix}T_{q}^{(J)}(f, g; a_{(J)}, b) \\ = \frac{b(1 − q)}{b − a_{(J)}} ∑_{i=0}^{N_{(J)}}(f(bq^{i}) − \frac{J_{q}(f; a_{(J)}, b)}{b − a_{(J)}})(g(bq^{i}) − \frac{J_{q}(g; a_{(J)}, b)}{b − a_{(J)}})q^{i}, \end{matrix}

where N_{(G)} = n − 1 and N_{(I)} = ∞. Under the conditions satisfied by the functions f and g, the product

\begin{matrix}(L(g(bq^{i}) − \frac{J_{q}(g; a_{(J)}, b)}{b − a_{(J)}}) − (f(bq^{i}) − \frac{J_{q}(f; a_{(J)}, b)}{b − a_{(J)}})) \\ ×((f(bq^{i}) − \frac{J_{q}(f; a_{(J)}, b)}{b − a_{(J)}}) − l(g(bq^{i}) − \frac{J_{q}(g; a_{(J)}, b)}{b − a_{(J)}})) \end{matrix}

is nonnegative for i = 0, 1, …, N_{(J)}, i.e.,

\begin{matrix}(f(bq^{i}) − \frac{J_{q}(f; a_{(J)}, b)}{b − a_{(J)}})^{2} + lL(g(bq^{i}) − \frac{J_{q}(g; a_{(J)}, b)}{b − a_{(J)}})^{2} \\ ≤ (l + L)(f(bq^{i}) − \frac{J_{q}(f; a_{(J)}, b)}{b − a_{(J)}})(g(bq^{i}) − \frac{J_{q}(g; a_{(J)}, b)}{b − a_{(J)}}). \end{matrix}

For J_{q}( ·; a_{(J)}, b) = G_{q}( ·; a, b), the desired inequality is obtained only by multiplying the upper inequalities by q^{i} and summing over i, i = 0, 1, …, n− 1. In the case J_{q}( ·; a_{(J)}, b) = I_{q}( ·; 0, b) it is needed to put n → ∞.

For J_{q}( ·; a_{(J)}, b) = R_{q}( ·; a, b) it should be noticed that under the conditions satisfied by f and g on [a, b] the inequality

l(ĝ(x) − I_{q}(ĝ; 0, 1)) ≤ f̂(x) − I_{q}(f̂; 0, 1) ≤ L(ĝ(x) − I_{q}(ĝ; 0, 1))

is valid for all x ∈ [0, 1]. Hence, according to (13) and the proved inequality for the previous case, we get desired inequality. □

REFERENCES

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[2] H. G a u c h m a n, Integral Inequalities in q-Calculus, Computers and Mathematics with Applications, vol. 47, (2004), 281–300.

[3] W. H a h n, ”Lineare Geometrische Differenzengleichungen”, 169 Berichte der Mathematisch-Statistischen Section im Forschungszentrum Graz, 1981.

[4] R. K o e k o e k, R. F. S w a r t t o u w, ”Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue”, Report of Delft University of Technology No 98–117, 1998.

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[6] P. M. R a j k o v i ć, M. S. S t a n k o v i ć, S. D. M a r i n k o v i ć, The zeros of polynomials orthogonal with respect to q-integral on several intervals in the complex plane, Proceedings of The Fifth International Conference on Geometry, Integrability and Quantization, 2003, Varna, Bulgaria (ed. I.M. Mladenov, A.C.Hirsshfeld), 178– 188.

M. S. Stanković Faculty of Occupational Safety University of Niš Serbia

e-mail: mstan@znrfak.ni.ac.yu

P. M. Rajković

Faculty of Mechanical Engineering University of Niš Serbia

                                          e-mail: pecar@masfak.ni.ac.yu

Sladjana D. Marinković Faculty of Electronic Engineering University of Niš Serbia

sladjana@elfak.ni.ac.yu