Theorem 1. Assume a convex function f : I⟶ℝ and g, h : [θ₁, θ₂]⟶ℝ are measurable functions such that g(θ) ∈ I and h(θ) ≥ 0∀θ ∈ [θ₁, θ₂]. Also, suppose that h, gh, (f∘g).h are all integrable functions on [θ₁, θ₂] and $$, then

Theorem 2. Assume a convex function f : [θ₁, θ₂]⟶ℝ, then the following double inequalities hold:

Theorem 3. Let 1 < p, q be such that 1/p + 1/q = 1 and assume two measurable functions say h, g : [θ₁, θ₂]⟶ℝ such that the functions |h(θ)|^p and |g(θ)|^q are integrable on [θ₁, θ₂]. Then

Theorem 4. Let f be a convex function defined on the interval I. Also, let h, u, v, w, z, g : [θ₁, θ₂]⟶ℝ be some integrable functions with the following conditions such as g(θ) ∈ I, h(θ), u(θ), v(θ), w(θ), z(θ) ∈ ℝ⁺ with u(θ) + w(θ) = 1, v(θ) + z(θ) = 1 for all θ ∈ [θ₁, θ₂] and $$. Then, for any nonempty subinterval $$ of [θ₁, θ₂] with $$ the following inequalities hold

For a concave function f, the reverse inequalities hold in (5).

Proof. Since u(θ) + w(θ) = 1, v(θ) + z(θ) = 1 for all θ ∈ [θ₁, θ₂], therefore for the subinterval $$ of [θ₁, θ₂] with $$, we have

Multiplying equation (6) by $$ and assigning it to the function f, then by convexity of f, we have

Also, by making use of the integral Jensen inequality, one has

From (7) and (8), we get (5).

Remark 5. The following is an equivalent form of the inequality (5)

Corollary 6. Let f be a convex function defined on [θ₁, θ₂]. Also, let u, v, w, z : [θ₁, θ₂]⟶ℝ be integrable functions such that u(θ), v(θ), w(θ), z(θ) ∈ ℝ⁺ with u(θ) + w(θ) = 1 and v(θ) + z(θ) = 1 for all θ ∈ [θ₁, θ₂]. Then, for a subinterval $$ of [θ₁, θ₂] with $$, the following inequalities hold

The direction of the inequalities reverses in (11), when the function f becomes concave.

Proof. Utilizing Theorem 4 for h(θ) = 1, g(θ) = θ for all θ ∈ [θ₁, θ₂], we get (11).

Corollary 7. If p, q ∈ ℝ and the functions u, v, w, z, h₁, g₁, and g₂ defined on [θ₁, θ₂] are nonnegative such that the functions $$ and wh₁g₁g₂, vh₁g₁g₂, zh₁g₁g₂, h₁g₁g₂ are integrable with u(θ) + w(θ) = 1 and v(θ) + z(θ) = 1 for all θ ∈ [θ₁, θ₂]. Also, if $$ is a subinterval of [θ₁, θ₂] with $$, then

• (A)

  For p, q > 1 such that (1/p) + (1/q) = 1, the following inequalities hold

• (B)

  For 0 < p < 1 and q = p/(p − 1) with $$, the following inequalities hold

• (C)

  For p < 0 and q = p/(p − 1) with $$, the inequalities in (13) hold

Proof.

• (A)

  In the case when p, q > 1, let $$. Then, by using Theorem 4 for f(θ) = θ^p, θ > 0, $$, and $$, we obtain (12). Also, let $$; then, applying the same procedure as above and replacing p, q, g₁, g₂ by q, p, g₂, g₁, respectively, we obtain (12)

For $$ and $$, the inequalities in (12) also hold. This can be proved as follows, since we know that

Taking integral, then with the proposed conditions, we obtain $$, which concludes the result.

• (B)

  In the case when 0 < p < 1, we have q < 0. So, applying (12) for p⟶1/p > 1, q⟶(1 − p)⁻¹, $$, and $$, we obtain (13)

• (C)

  In the case when p < 0, we have 0 < q < 1. So, applying the arguments of part B with replacing p, q, g₁(θ), g₂(θ) by q, p, g₂(θ), g₁(θ), respectively, we get (13)

Corollary 8. Let p, q ∈ ℝ and u, v, w, z, h₁, g₁, g₂ be nonnegative functions defined on [θ₁, θ₂] such that $$ and $$ are integrable functions with u(θ) + w(θ) = 1 and v(θ) + z(θ) = 1 for all θ ∈ [θ₁, θ₂]. Also, assume that $$ is a subinterval of [θ₁, θ₂] with $$, then

• (A)

  For p > 1, q = p/(p − 1) and $$, the following inequalities hold

• (B)

  For 0 < p < 1 and q = p/(p − 1) with $$, the following inequalities hold

• (C)

  For p < 0 and q = p/(p − 1) with $$, the result in (16) holds

Proof.  

• (A)

  Let f(θ) = θ^1/p, θ > 0 which is clearly a concave function for p > 1. Thus, by using Theorem 4 for f(θ) = θ^1/p, $$, and $$, we obtain (15). If $$, then adopting the same procedure and replacing p, q, g₁, g₂ by q, p, g₂, g₁, respectively, we obtain (15)

For $$ and $$, the inequalities in (12) also hold. This can be proved as follows, since we know that

Hence, taking integral and using the proposed conditions, we get $$, which verifies the result.

• (B)

  In the case when 0 < p < 1, applying (15) for p⟶1/p > 1, q⟶(1 − p)⁻¹, $$, and $$, we obtain (16)

• (C)

  In the case when 0 > p, we have q ∈ (0, 1), which shows that this case reflects case B; therefore, applying the arguments of case B with replacing p, q, g₁(θ), g₂(θ) by q, p, g₂(θ), g₁(θ), respectively, we get (16)

Corollary 9. Assume some positive integrable functions h, u, v, w, z and g defined on [θ₁, θ₂] with u(θ) + w(θ) = 1 and v(θ) + z(θ) = 1 for θ ∈ [θ₁, θ₂]. Also, assume that α, β ∈ ℝ such that α ≤ β, then

• (A)

  For α/β ∈ ℝ − {(−1, 0)}, β ≠ 0, the following inequalities hold

• (B)

  For β/α ∈ ℝ − {(−1, 0)}, α ≠ 0, the following inequalities hold

Proof.  

• (A)

  Let f(θ) = θ^α/β for θ > 0, then the following possible cases can be discussed:

Case 1. If α, β ∈ ℝ⁻ with α ≤ β, then α/β ≥ 1, and the function f(θ) is convex. Therefore, utilizing (5) for f(θ) and g(θ)⟶g^β(θ), after that taking power 1/α, we obtain (20)

Case 2. If α, β ∈ ℝ⁺ with α < β, then 0 < α/β < 1, and the function f(θ) is concave. Therefore, utilizing (5) for f(θ) and g(θ)⟶g^β(θ) and then taking power 1/α, we obtain (20)

Case 3. If β ∈ ℝ⁺, α ∈ ℝ⁻ with α ≤ β, then α/β ≤ 0, and the function f(θ) is convex. Therefore, utilizing (5) for f(θ) and g(θ)⟶g^β(θ) and then taking power 1/α, we obtain (20)

For the case when α = 0, taking $$ of (20), we get (21).

• (B)

  Let f(θ) = θ^β/α for θ > 0, then the following possible cases can be discussed:

Case 1. If α, β ∈ ℝ⁺ with α ≤ β, then β/α ≥ 1, and the function f(θ) is convex. Hence, using (5) for f(θ) and g(θ)⟶g^α(θ) and then taking power 1/β, we obtain (22)

Case 2. If α, β ∈ ℝ⁻ with α ≤ β, then 0 < β/α < 1, and the function f(θ) is concave. Hence, using (5) for f(θ) and g(θ)⟶g^α(θ) and then taking power 1/β, we obtain (22)

Case 3. Similarly, if α ∈ ℝ⁻, β ∈ ℝ⁺ with α ≤ β, then β/α ≤ −1, and the function f(θ) is convex function. Hence, using (5) for f(θ) and g(θ)⟶g^α(θ) and taking power 1/β, we obtain (22)

For the case when β = 0, taking lim_β⟶0 in (22), we get (23).

Corollary 10. Assume some positive integrable functions h, u, v, w, z defined on [θ₁, θ₂] such that u(θ) + w(θ) = 1 and v(θ) + z(θ) = 1 for θ ∈ [θ₁, θ₂] and further assume that g is an arbitrary integrable function defined on [θ₁, θ₂]. Also, suppose that p is a strictly monotone continuous function whose domain is the image of g. Then, for (Ψ∘p⁻¹)(θ) as a convex function, the following inequalities hold

The direction of the inequalities reverses in (26), when the function (Ψ∘p⁻¹)(θ), becomes concave.

Proof. The desired inequalities can be calculated by utilizing (5) for g⟶p∘g and f⟶Ψ∘p⁻¹.

Definition 11 (Csiszâr divergence [25]). Assume that Ψ : I⁺⟶ℝ is a function defined on a positive interval I⁺. Also, assume that p, q : [θ₁, θ₂]⟶(0, ∞) are two integrable functions such that q(θ)/p(θ) ∈ I⁺ for all θ ∈ [θ₁, θ₂], then the integral form of Csiszâr-divergence is defined by

Theorem 12. Let Ψ : I⁺⟶ℝ be a convex function defined on a positive interval I⁺ and assume that u, v, w, z, p, q : [θ₁, θ₂]⟶ℝ⁺ are integrable functions such that q(θ)/p(θ) ∈ I⁺, u(θ) + w(θ) = 1, and v(θ) + z(θ) = 1 for all θ ∈ [θ₁, θ₂]. Then, for a subinterval $$ of [θ₁, θ₂] with $$, the following inequalities hold

Proof. Using Theorem 4 for f⟶Ψ, g(θ) = q(θ)/p(θ), and h(θ) = p(θ), we obtain (28).

Definition 13 (Shannon entropy). The Shannon entropy for a positive probability density function p(θ) defined on [θ₁, θ₂] is given by

Corollary 14. Let u, v, w, z : [θ₁, θ₂]⟶ℝ⁺ be integrable functions such that u(θ) + w(θ) = 1 and v(θ) + z(θ) = 1 for all θ ∈ [θ₁, θ₂]. Also, let p, q be two positive probability density functions defined on [θ₁, θ₂], then for a subinterval $$ of [θ₁, θ₂] with $$, the following inequalities hold

Proof. Using (28) for the function Ψ(ζ) = −logζ, ζ ∈ I⁺, we obtain (30).

Remark 15. For q(θ) = 1, the result (30) becomes

Definition 16 (Kullback-Leibler divergence). The Kullback-Leibler divergence for two positive probability densities p(θ) and q(θ) defined on [θ₁, θ₂] is given by

Corollary 17. Let u, v, w, z, p, q : [θ₁, θ₂]⟶ℝ⁺ be integrable functions such that p(θ) and q(θ) are positive probability density functions and u(θ) + w(θ) = 1 and v(θ) + z(θ) = 1 for all θ ∈ [θ₁, θ₂]. Then, for a subinterval $$ of [θ₁, θ₂] with $$ the following inequalities hold

Proof. Using (28) for the convex function Ψ(ζ) = ζlogζ, ζ ∈ I⁺, we obtain (33).

Definition 18 (variational distance). The variational distance for two positive probability densities p(θ) and q(θ) defined on [θ₁, θ₂] is given by

Corollary 19. Let u, v, w, z, p, q and $$ be interpreted as in Corollary 17, then

Proof. Using the convex function Ψ(ζ) = |ζ − 1| for ζ ∈ I⁺ in (28), we obtain (35).

Definition 20 (Jeffrey’s distance). Jeffrey’s distance for two positive probability density functions p(θ) and q(θ) defined on [θ₁, θ₂] is given by

Corollary 21. Let u, v, w, z, p, q and $$ be interpreted as in Corollary 17, then

Proof. Using the convex function Ψ(ζ) = (ζ − 1)logζ, ζ ∈ I⁺ in (28), we obtain (37).

Definition 22 (Bhattacharyya coefficient). The Bhattacharyya coefficient for two positive probability density functions p(θ) and q(θ) defined on [θ₁, θ₂] is given by

Corollary 23. Let u, v, w, z, p, q and $$ be interpreted as in Corollary 17, then

Proof. Using the convex function $$, ζ ∈ I⁺ in (28), we obtain (39).

Definition 24 (Hellinger distance). The Hellinger distance for two positive probability density functions p(θ) and q(θ) defined on [θ₁, θ₂] is given by

Corollary 25. Let u, v, w, z, p, q and $$ be defined as in Corollary 17, then

Proof. Using (28) for the convex function $$ we obtain (41).

Definition 26 (triangular discrimination). The triangular discrimination for two positive probability density functions p(θ) and q(θ) defined on [θ₁, θ₂] is given by

Corollary 27. Let u, v, w, z, p, q and $$ be as stated in Corollary 17, then

Proof. Utilizing the convex function Ψ(ζ) = (ζ − 1)²/(ζ + 1), ζ ∈ I⁺ in (28), we obtain (43).