Milne and Hermite-Hadamard's type inequalities for strongly multiplicative convex function via multiplicative calculus

Muhammad Umar; Saad Ihsan Butt; Youngsoo Seol; Muhammad Umar; Saad Ihsan Butt; Youngsoo Seol

doi:10.3934/math.20241625

AIMS Mathematics

2024, Volume 9, Issue 12: 34090-34108. doi: 10.3934/math.20241625

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Milne and Hermite-Hadamard's type inequalities for strongly multiplicative convex function via multiplicative calculus

1.
Department of Mathematics, COMSATS University Islamabad, Lahore Campus, Pakistan
2.
Department of Mathematics, Dong-A University, Busan 49315, Korea

Received: 24 September 2024 Revised: 13 November 2024 Accepted: 22 November 2024 Published: 03 December 2024
MSC : 26D10, 26D15, 26E60, 90C23

In this paper, we take into account the notion of strongly multiplicative convex function and derive integral inequalities of Hermite-Hadamard ($ H.H $) type for such a function in the frame of multiplicative calculus. We also develop integral inequalities of $ H.H $ type for product and quotient of strongly multiplicative convex and strongly multiplicative concave functions via multiplicative calculus. All the results of the theorems are verified graphically by taking into account some reasonable examples. Additionally, we establish the inequalities of the Milne type for strongly multiplicative convex functions.
- convex function,
- multiplicative convex function,
- strongly multiplicative convex function,
- Hermite-Hadamard type inequality,
- Milne type inequality
Citation: Muhammad Umar, Saad Ihsan Butt, Youngsoo Seol. Milne and Hermite-Hadamard's type inequalities for strongly multiplicative convex function via multiplicative calculus[J]. AIMS Mathematics, 2024, 9(12): 34090-34108. doi: 10.3934/math.20241625

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Abstract

In this paper, we take into account the notion of strongly multiplicative convex function and derive integral inequalities of Hermite-Hadamard ($ H.H $) type for such a function in the frame of multiplicative calculus. We also develop integral inequalities of $ H.H $ type for product and quotient of strongly multiplicative convex and strongly multiplicative concave functions via multiplicative calculus. All the results of the theorems are verified graphically by taking into account some reasonable examples. Additionally, we establish the inequalities of the Milne type for strongly multiplicative convex functions.

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