Page 168 - IJOCTA-15-1

J. Khan, M. Adil Khan, S. Sarwar / IJOCTA, Vol.15, No.1, pp.155-165 (2025)
ακ(v − u) ακ(v − u)
≤ 2 N 1 (α, κ, p) ≤ 2 N 1 (α, κ, p)
(α + κ) (α + κ)

1
q
q

× ′ ′ q

N 2 (α, κ) ω (u) + N 3 (α, κ) ω (v)
q
q
′
′

× N 2 (α, κ) ω (u) + N 3 (α, κ) ω (v) 1 q

!
1
q
q
′
′

+ L 4 (α, κ) ω (u) + L 5 (α, κ) ω (v) q .

!
1
q
′
′

+ L 4 (α, κ) ω (u) + L 5 (α, κ) ω (v) q q Remark 4. Substituting α = κ = 1, u = δ 1 and

v = δ 2 in (18), we obtain
(18) Z δ 2
1 ω(δ 1 ) + ω(δ 2 )
holds, where ω(s)ds −
2
δ 2 − δ 1
δ 1
q
1  q  1
Z ′ ′ q
1 α p p α
N 1 (α, κ, p) = t κ −1 dt , N 2 (α, κ) = , ≤ δ 2 − δ 1 ω (δ 1 ) + 3 ω (δ 2 )
0 2(α + κ) 4  4 
q
α(α + 2κ) 2α + κ  ′ q ′  1 q !
N 3 (α, κ) = , N 4 (α, κ) = , 3 ω (δ 1 ) + 2 ω (δ 2 )

2(α + κ) 2(α + κ)
+   .
4
κ
N 5 (α, κ) = .
α + κ Acknowledgments
None.
Proof. Applying H¨older’s inequality in (17), we Funding
have
None.
α
−1 2
(α + κ) κ Γ κ (α) α α,κ
α J Conflict of interest
α αu+κv + ω(v)
κ κ ( )
(v − u) κ α+κ
The authors declare that they have no conflict of

κ α,κ αω(u) + κω(v)
+ α J αu+κv − ω(u) − interest regarding the publication of this article.
α κ −1 ( α+κ ) α + κ
ακ(v − u) Author contributions
≤
(α + κ) 2
Conceptualization: Muhammad Adil Khan
1 q 1
Z
1 p αt αt q Formal analysis: Sana Sarwar
α p ′
t κ −1 dt ω u + (1 − )v dt
α + κ α + κ Methodology: Muhammad Adil Khan
0
1 q 1 Supervision: Muhammad Adil Khan
Z
1 α p p κt κt q
ω
+ t κ −1 dt ′ (1 − )u + ( )v dt . Writing – original draft: Jamroz Khan
α + κ α + κ
0
Writing – review & editing: Sana Sarwar
′ q
Since |ω | is convex, therefore
α Availability of data
−1 2
(α + κ) κ Γ κ (α) α α,κ
α J αu+κv + ω(v)
α
(v − u) κ α+κ
κ κ ( ) Not applicable.

κ α,κ αω(u) + κω(v)
+ α J αu+κv − ω(u) − References
α κ −1 ( α+κ ) α + κ
1 [1] Peˇcari´c, J., Proschan, F. & Tong, Y. L. (1992).
ακ(v − u) 1 α p p
Z
≤ t κ −1 dt Convex Functions, Partial Orderings, and Statis-
(α + κ) 2 0 tical Applications. Academic Press Inc. https:
1
1 1
Z Z //doi.org/10.1016/s0076-5392%2808%29x61
αt q αt q q
′
′
dt ω (u) + (1 − )dt ω (v) 62-4

α + κ α + κ
0 0 [2] Zhao, T. H., Wang, M. K., & Chu, Y. M. (2021).
Z 1 κt Monotonicity and convexity involving generalized
′
+ (1 − )dt ω (u) q elliptic integral of the first kind. Revista de la Real
0 α + κ
1 ! Academia de Ciencias Exactas, Fysicas y Natu-
Z 1 κt q q
′
+ dt ω (v) rales (Espana), 115(2), 1-13. http://dx.doi.o
0 α + κ rg/10.1007/s13398-020-00992-3
162