Definition 1 (p-convex set see [10, 11]). A set M = [d₁, d₂]⊆ℝ\{0} is p -convex set, if

for all x, y ∈ M, j ∈ [0, 1], where p = 2u + 1 or p = a/b, a = 2v + 1, b = 2w + 1 and u, v, w ∈ N.

Definition 2 (p-convex function see [10]). Let M = [d₁, d₂]⊆ℝ\{0} be a p -convex set. A function ψ : M = [d₁, d₂]⟶ℝ is called p-convex function, if

holds for all x, y ∈ M and j ∈ [0, 1].

Definition 3 (Strongly convex function see [4]). A function ψ : M = [d₁, d₂]⟶ℝ is called strongly convex function with modulus μ on M, where μ ≥ 0, if

holds for all x, y ∈ M and j ∈ [0, 1].

Definition 4 (Strongly p-convex function see [12]). A function ψ : M = [d₁, d₂]⟶ℝ is called strongly p-convex function, if

holds for all x, y ∈ M and j ∈ [0, 1].

Definition 5 (Harmonic convex set see [11, 13]). A set M = [d₁, d₂]⊆ℝ\{0} is said to be harmonic convex set, if

for all x, y ∈ M and j ∈ [0, 1].

Definition 6 (Harmonic convex function see [11, 14]). Let M = [d₁, d₂]⊆ℝ\{0} be the harmonic convex set. A function ψ : M = [d₁, d₂]⊆ℝ\{0}⟶ℝ is harmonic convex function, if

holds for all x, y ∈ M and j ∈ [0, 1].

Definition 7 (p-harmonic convex set see [11, 15]). A set M = [d₁, d₂]⊆ℝ\{0} is p -harmonic convex set, if

for all x, y ∈ M and j ∈ [0, 1].

Definition 8 (p-harmonic convex function see [11, 15]). Let M = [d₁, d₂]⊆ℝ\{0} be p-harmonic convex set. A function ψ : M = [d₁, d₂]⊆ℝ\{0}⟶ℝ is p-harmonic convex, if

holds for all x, y ∈ M and j ∈ [0, 1].

Definition 9 (Strongly reciprocally convex function see [16]). Let M be an interval and let μ ∈ (0, ∞). A function ψ : M = [d₁, d₂]⊆ℝ\{0}⟶ℝ is said to be strongly reciprocally convex function with modulus μ on M, if

holds for all x, y ∈ M and j ∈ [0, 1].

Definition 10 (Strongly reciprocally p-convex function see [17]). Let M be a p -harmonic convex set and let μ ∈ (0, ∞). A function ψ : M = [d₁, d₂]⊆ℝ\{0}⟶ℝ is said to be strongly reciprocally p-convex function with modulus μ on M, if

holds for all x, y ∈ M and j ∈ [0, 1].

Definition 11 (h-convex function see [18]). Choose the functions ψ, h : M = [d₁, d₂]⟶ℝ that are non-negative; then, ψ is called h-convex function, if

for all x, y ∈ M and j ∈ [0, 1].

Definition 12 (Strongly reciprocally (p, h)-convex function). Let M be a p-harmonic convex set and let μ ∈ (0, ∞). A function ψ : M = [d₁, d₂]⊆ℝ\{0}⟶ℝ is said to be strongly reciprocally (p, h)-convex function with modulus μ on M, if

hold for all x, y ∈ M and j ∈ [0, 1].

Throughout the paper, for convenience, we represent the class of strongly reciprocally (p, h) convex functions by SR(p, h).

Remark 13. Inserting h(j) = j in Definition 12, we obtain Definition 10, and inserting h(j) = j and p = 1, Definition 12 reduces to Definition 9.

Similarly, from Definition 12, Definitions 6 and 8 can be obtained by inserting h(j) = j, μ = 0, p = 1, and h(j) = j with μ = 0, respectively.

Proposition 14. Let ψ, σ : M⟶ℝ be two strongly reciprocally (p,h)-convex functions with modulus μ on M; then, ψ + σ : M⟶ℝ is also strongly reciprocally (p,h)-convex function with modulus μ^∗ on M, where μ^∗ = 2μ.

Proof. We will start by definition of strongly reciprocally (p,h)-convexity of ψ and σ:

which in turns implies that

where μ^∗ = 2μ and μ ≥ 0. This completes the proof.

Proposition 15. Let ψ : M⟶ℝ be a strongly reciprocally (p,h)-convex function; then, for any λ ≥ 0, λψ : M⟶ℝ is also strongly reciprocally (p,h)-convex function with modulus ν^∗ on M, where ν^∗ = λμ.

Proof. Let λ ≥ 0, ψ ∈ SR(p, h), we obtain

where ν^∗ = λμ and μ ≥ 0. This completes the proof.

Proposition 16. Let ψ_i : M⟶ℝ, where 1 ≤ i ≤ n be in SR(p, h) with modulus μ; then, for λ_i ≥ 0 where 1 ≤ i ≤ n, the function ψ : M⟶ℝ, where $$ is also in SR(p, h) with modulus γ ≥ 0, where $$.

Proof. Let M is a p-harmonic convex set. Then, ∀x, y ∈ M and j ∈ [0, 1], we have

where $$. This completes the proof.

Proposition 17. Let ψ_i : M⟶ℝ, where 1 ≤ i ≤ n be strongly reciprocally (p,h)-convex functions with modulus μ; then ψ = max{ψ_i, i = 1, 2, ⋯, n}, is also strongly reciprocally (p,h)-convex functions with modulus μ.

Proof. Let M is p-harmonic convex set. Then ∀x, y ∈ M and j ∈ [0, 1], we have

This completes the proof.

Theorem 18. Let M ⊂ ℝ\{0} be an interval. If ψ : M⟶ℝ be a strongly reciprocally (p,h)-convex function with modulus μ ≥ 0 and ψ ∈ L[d₁, d₂], then for h(1/2) ≠ 0, we have

Proof. Since, ψ ∈ SR(p, h), and allowing j = 1/2 yields

Let $$ and $$ and integrating (21) w.r.t j over [0, 1], we obtain

and

which gives one side of (20).

For the right side of (20), since ψ ∈ SR(p, h), so by setting x = d₁ and y = d₂, we obtain the following result:

Integrating (24) w.r.t j over [0, 1], we obtain

and

which is right side of (20), which completes the proof.

Remark 19.

• (1)

  Inserting h(j) = j and p = 1 in Theorem 18, we obtained Hermite-Hadamard inequality for strongly reciprocally convex function [16] (Theorem 3.1)

• (2)

  Insertion h(j) = j, p = 1, and μ = 0 in Theorem 18 yields Hermite-Hadamard inequality for harmonic convex functions [14] (Theorem 2.4)

Now, we develop Fejér type inequality for this new class of convex functions.

Theorem 20. Let M ⊂ ℝ\{0} be an interval. If ψ : M⟶ℝ be a strongly reciprocally (p,h)-convex function with modulus μ ≥ 0, then for h(1/2) ≠ 0, we have

holds for d₁, d₂ ∈ M with d₁ ≤ d₂ and ψ ∈ L[d₁, d₂], where w : M⟶ℝ is a non-negative integrable function that satisfies

Proof. Since ψ ∈ SR(p, h), and allowing j = 1/2 yields

Inserting $$ and $$ in (29), we obtain

Since, w is a non-negative symmetric and integrable function, so

Integrating inequality (31) w.r.t j over [0, 1], we obtain

and

so,

which is left side of (27).

Remark 21.

• (1)

  Inserting h(j) = j and p = 1 in Theorem 20, we obtained Fejér type inequality for strongly reciprocally convex function [16] (Theorem 3.7)

• (2)

  Insertion h(j) = j, in Theorem 20, yields Fejér type inequality for strongly reciprocally p-convex functions [17] (Theorem 3.5)

Lemma 22 see ([23], Lemma 2.1). Let ψ : M = [d₁, d₂]⊆ℝ is differentiable defined on the interior M of M. If ψ^′ ∈ L[d₁, d₂] and λ ∈ [0, 1], then

As a first application of Lemma 22, we prove the following result.

Theorem 23. Let M = [d₁, d₂] ⊂ ℝ\{0} be a p -harmonic convex set, and let ψ : M = [d₁, d₂]⊆ℝ\{0}⟶ℝ be differentiable defined on the interior M of M. If M^′ ∈ L[d₁, d₂] and $$ is strongly reciprocally (p,h)-convex function on M, q ≥ 1, and λ ∈ [0, 1], then

where

Proof. Using Lemma 22 and power mean inequality, we have

Since $$, so

This completes the proof.

Remark 24. Inserting μ = 0 and h(j) = j in Theorem 23, we obtained [23] Theorem 2.2.

For q = 1, Theorem 23 reduces to the following result.

Corollary 25. Let M = [d₁, d₂] ⊂ ℝ\{0} be a p -harmonic convex set, and let ψ : M = [d₁, d₂]⊆ℝ\{0}⟶ℝ be differentiable defined on the interior M of M. If ψ^′ ∈ L[d₁, d₂] and $$ is in SR(p, h) on M and λ ∈ [0, 1], then

where k₁₅, k₁₆, k₁₇, k₁₈, k₇, and k₈ are given by (43) to (48).

Remark 26. Inserting h(j) = j and μ = 0 in Corollary 25, we obtained [23] Corollary 2.3.

Using Lemma 22, we can prove the following result.

Theorem 27. Let M = [d₁, d₂] ⊂ ℝ\{0} be a p -harmonic convex set, and let ψ : M = [d₁, d₂]⊆ℝ\{0}⟶ℝ be differentiable defined on the interior M of M. If M^′ ∈ L[d₁, d₂] and $$ is in SR(p, h) on M, r, q > 1, (1/r) + (1/q) = 1 and λ ∈ [0, 1], then

where

Proof. Using Lemma 22, we have

Applying Holder’s integral inequality,

Since $$, so

Which is the required result.

Remark 28. Inserting h(j) = j and μ = 0 in Theorem 27, we obtained [23] Theorem 2.5.

For λ = 0, Theorem 27 reduces to the following result.

Corollary 29. Let M = [d₁, d₂] ⊂ ℝ\{0} be a p-harmonic convex set, and let ψ : M = [d₁, d₂]⊆ℝ\{0}⟶ℝ be differentiable defined on the interior M of M. If ψ^′ ∈ L[d₁, d₂] and $$ is in SR(p, h) on M, r, q > 1, (1/r) + (1/q) = 1 and λ ∈ [0, 1], then

where k₁₉, k₂₀, k₂₁, k₂₂, k₁₃, and k₁₄ are given by (53)-(58).

Remark 30. Inserting h(j) = j and μ = 0 in Corollary 29, we obtained [23] Corollary 3.5.

For λ = 1, Theorem 27 reduces to the following result.

Corollary 31. Let M = [d₁, d₂] ⊂ ℝ\{0} be a p -harmonic convex set, and let ψ : M = [d₁, d₂]⊆ℝ\{0}⟶ℝ be differentiable defined on the interior M of M. If M^′ ∈ L[a₁, b₁] and $$ is in SR(p, h) on M, r, q > 1, (1/r) + (1/q) = 1 and λ ∈ [0, 1] then,

where k₁₉, k₂₀, k₂₁, k₂₂, k₁₃, and k₁₄ are given by (53)-(58).

Remark 32. Inserting h(j) = j and μ = 0 in Corollary 31, we obtained [23] Corollary 3.6.

For λ = 1/3, Theorem 27 reduces to the following result.

Corollary 33. Let M = [d₁, d₂] ⊂ ℝ\{0} be a p-harmonic convex set, and let ψ : M = [d₁, d₂]⊆ℝ\{0}⟶ℝ be differentiable defined on the interior M of M. If ψ^′ ∈ L[d₁, d₂] and $$ is in SR(p, h) on M, r, q > 1, (1/r) + (1/q) = 1 and λ ∈ [0, 1] then,

where k₁₉, k₂₀, k₂₁, k₂₂, k₁₃, and k₁₄ are given by (53)-(58).

Remark 34. Inserting h(j) = j and μ = 0 in Corollary 33, we obtained [23] Corollary 3.7.

For λ = 1/2, Theorem 27) reduces to the following result.

Corollary 35. Let M = [d₁, d₂] ⊂ ℝ\{0} be a p-harmonic convex set, and let ψ : M = [d₁, d₂]⊆ℝ\{0}⟶ℝ be differentiable defined on the interior M of M. If ψ^′ ∈ L[d₁, d₂] and $$ is in SR(p, h) on M, r, q > 1, (1/r) + (1/q) = 1 and λ ∈ [0, 1] then,

where k₁₉, k₂₀, k₂₁, k₂₂, k₁₃, and k₁₄ are given by (53)–(58).

Remark 36. Inserting h(j) = j and μ = 0 in Corollary 35, we obtained [23] Corollary 3.8.