Research article

Bounds for certain function related to the incomplete Fox-Wright function

  • Received: 23 January 2024 Revised: 22 April 2024 Accepted: 28 April 2024 Published: 07 June 2024
  • MSC : 25D15, 26D15, 30C45, 30D15, 33E20

  • Motivated by the recent investigations of several authors, the main aim of this article is to derive several functional inequalities for a class of functions related to the incomplete Fox-Wright functions that were introduced and studied recently. Moreover, new functional bounds for functions related to the Fox-Wright function are deduced. Furthermore, a class of completely monotonic functions related to the Fox-Wright function is given. The main mathematical tools used to obtain some of the main results are the monotonicity patterns and the Mellin transform for certain functions related to the two-parameter Mittag-Leffler function. Several potential applications for this incomplete special function are mentioned.

    Citation: Khaled Mehrez, Abdulaziz Alenazi. Bounds for certain function related to the incomplete Fox-Wright function[J]. AIMS Mathematics, 2024, 9(7): 19070-19088. doi: 10.3934/math.2024929

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  • Motivated by the recent investigations of several authors, the main aim of this article is to derive several functional inequalities for a class of functions related to the incomplete Fox-Wright functions that were introduced and studied recently. Moreover, new functional bounds for functions related to the Fox-Wright function are deduced. Furthermore, a class of completely monotonic functions related to the Fox-Wright function is given. The main mathematical tools used to obtain some of the main results are the monotonicity patterns and the Mellin transform for certain functions related to the two-parameter Mittag-Leffler function. Several potential applications for this incomplete special function are mentioned.



    The Fox-Wright function, denoted by pΨq, which is a generalization of hypergeometric functions, and it defined as follows [1] (see also [2, p. 4, Eq (2.4)]):

    pΨq[(a1,A1),,(ap,Ap)(b1,B1),,(bq,Bq)|z]=pΨq[(ap,Ap)(bq,Bq)|z]=k=0pl=1Γ(al+kAl)ql=1Γ(bl+kBl)zkk!, (1.1)

    where Aj0(j=1,,p) and Bl0(l=1,,q). The convergence conditions and convergence radius of the series on the right-hand side of (1.1) immediately follow from the known asymptote of the Euler gamma function. The defining series in (1.1) converges in the complex z-plane when

    Δ=1+qj=1Bjpi=1Ai>0.

    If Δ=0, then the series in (1.1) converges for |z|<ρ and |z|=ρ under the condition (μ)>12, where

    ρ=(pi=1AAii)(qj=1BBjj),μ=qj=1bjpk=1ak+pq2.

    The Fox-Wright function extends the generalized hypergeometric function pFq[z] the power series form of which is as follows [3, p. 404, Eq (16.2.1)]:

    pFq[apbq|z]=k=0pl=1(al)kql=1(bl)kzkk!,

    where, as usual, we make use of the Pochhammer symbol (or rising factorial) given below:

    (τ)0=1;(τ)k=τ(τ+1)(τ+k1)=Γ(τ+k)Γ(τ),kN.

    In the special case that Ap=Bq=1 the Fox-Wright function pΨq[z] reduces (up to the multiplicative constant) to the following generalized hypergeometric function:

    pΨq[(ap,1)(bq,1)|z]=Γ(a1)Γ(ap)Γ(b1)Γ(bq)pFq[apbq|z].

    For p=q=a1=A1=1,b1=β, and B1=α, we recover from (1.1) the two-parameter Mittag-Leffler function Eα,β(z) (also known as the Wiman function [4]) defined as follows (see, for example, [5, Chapter 4]):

    Eα,β(t)=k=0tkΓ(αk+β),min(α,β,t)>0. (1.2)

    To provide the exposition of the results in the present investigation, we need the so-called incomplete Fox-Wright functions pΨ(γ)q[z] and pΨ(γ)q[z] that were introduced by Srivastava et al. in [6, Eqs (6.1) & (6.6)]:

    pΨ(γ)q[(μ,M,x),(ap1,Ap1)(bq,Bq)|z]=k=0γ(μ+kM,x)p1j=1Γ(aj+kAj)qj=1Γ(bj+kBj)zkk!,

    and

    pΨ(Γ)q[(μ,M,x),(ap1,Ap1)(bq,Bq)|z]=k=0Γ(μ+kM,x)p1j=1Γ(aj+kAj)qj=1Γ(bj+kBj)zkk!,

    where γ(a,x) and Γ(a,x) denote the lower and upper incomplete gamma functions, the integral expression of which is as follows [3, p. 174, Eq (8.2.1-2)]):

    γ(ν,x)=x0ettν1dt,x>0,(ν)>0, (1.3)

    and

    Γ(ν,x)=xettν1dt,x>0,(ν)>0. (1.4)

    These two functions satisfy the following decomposition formula [3, p. 136, Eq (5.2.1)]:

    Γ(ν,x)+γ(ν,x)=Γ(ν),(ν)>0. (1.5)

    The positivity constraint of parameters M,Aj,Bj>0 should satisfy the following constraint:

    Δ(γ)=1+qj=1BjMp1i=1Ai0,

    where the convergence conditions and characteristics coincide with the ones around the 'complete' Fox-Wright function pΨq[z].

    The properties of some functions related to the incomplete special functions including their functional inequalities, have been the subject of several investigations [7,8,9,10,11,12,13]. A certain class of incomplete special functions are widely used in some areas of applied sciences due to the relations with well-known and less-known special functions, such as the Nuttall Q-function [14], the generalized Marcum Q-function (see e.g., [14, p. 39]), the McKay Iν Bessel distribution (see e.g., [15, Theorem 1]), the McKay Kν(a,b) distribution [16], and the non-central chi-squared distribution [17, Section 5]. The incomplete Fox-Wright functions have important applications in communication theory, probability theory, and groundwater pumping modeling; see [6, Section 6] for details. See also [18]. To date, there have been many studies on a some class of functions related to the lower incomplete Fox-Wright functions; see, for instance [19,20,21]. Also, Mehrez et al. [22] considered a new class of functions related to the upper incomplete Fox-Wright function, defined in the following form:

    K(ν)α,β(a,b)=2ν12ea222Ψ(Γ)1[(ν+12,12,b22),(1,1)(β,α)|a2], (1.6)
    (a>0,b0,α12,β>0,ν>1).

    In [22], several properties of the function defined by (1.6), including its differentiation formulas, fractional integration formulas that can be obtained via fractional calculus and new summation formulas that comprise the incomplete gamma function, as well as some other special functions (such as the complementary error function) are investigated. In this paper, we apply another point of view to the following upper incomplete Fox-Wright function:

    2Ψ(Γ)1[(ν,12,b22),(1,1)(β,α)|z]=21νez22K(2ν1)α,β(z2,b), (1.7)
    (min(z,ν,β)>0,b0,α12).

    By using certain properties of the two parameters of the Mittag-Leffler and incomplete gamma functions, we derive new functional inequalities based on the aforementioned function defined in (1.7). Furthermore, two classes of completely monotonic functions are presented.

    Before proving our main results, we need the following useful lemmas. One of the main tools is the following result, i.e., which entails applying the Mellin transform on [b,) of the function et22Eα,β(t):

    Lemma 2.1. [22] The following integral representation holds true:

    2Ψ(Γ)1[(ν,12,b22),(1,1)(β,α)|z]=21νbt2ν1et22Eα,β(zt2)dt, (2.1)
    (min(z,ν,β)>0,b0,α12).

    Remark 2.2. If we set b=0 in Lemma 2.1, we obtain

    2Ψ1[(ν,12),(1,1)(β,α)|z]=21ν0t2ν1et22Eα,β(zt2)dt, (2.2)
    (z>0,α12,β>0,ν>1).

    Lemma 2.3. [23] Let (ak)k0 and (bk)k0 be two sequences of real numbers, and let the power series f(t)=k=0aktk and g(t)=k=0bktk be convergent for |t|<r. If bk>0 for k0 and if the sequence (ak/bk)k0 is increasing (decreasing) for all k, then the function tf(t)/g(t) is also increasing (decreasing) on (0,r).

    The following lemma, is one of the crucial facts in the proof of some of our main results.

    Lemma 2.4. If min(α,β)>1, then the function tetEα,β(t) is decreasing on (0,).

    Proof. From the power-series representations of the functions tEα,β(t) and tet, we get

    etEα,β(t)=(k=0tkΓ(αk+β))/(k=0tkΓ(k+1))=:(k=0aktk)/(k=0bktk).

    Given lemma 2.3, to prove that the function tetEα,β(t) is decreasing, it is sufficient to prove that the sequence (ck)k0=(ak/bk)k0 is decreasing. A simple computation gives

    ck+1ck=Γ(k+2)Γ(αk+β)Γ(k+1)Γ(αk+α+β),k0. (2.3)

    Moreover, since the digamma function ψ(t)=Γ(t)/Γ(t) is increasing on (0,), we get that the function

    tΓ(t+λ)Γ(t),λ>0,

    is increasing on (0,). This implies that the inequality

    Γ(t+λ)Γ(t+δ)Γ(t)Γ(t+λ+δ), (2.4)

    holds true for all λ,δ>0. Now, we set t=k+1,λ=1 and δ=(α1)k+β1 in (2.4), we get

    Γ(αk+β)Γ(k+2)Γ(k+1)Γ(αk+β+1). (2.5)

    Using the fact that Γ(αk+α+β)>Γ(αk+β+1) for all min(α,β)>1, and in consideration of (2.5), we obtain

    Γ(αk+β)Γ(k+2)Γ(k+1)Γ(αk+β+α). (2.6)

    Bearing in mind (2.3) and the inequality (2.6), we can show that the sequence (ck)k0 is decreasing. This, in turn, implies that the function tetEα,β(t) is decreasing on (0,) for all min(α,β)>1.

    Lemma 2.5. Let α>0 and β>0. If

    (α,β)J:={(α,β)R2+:Γ(β)Γ2(α+β)<2Γ(2α+β)<1Γ(α+β)}, (2.7)

    then

    eηα,βtΓ(β)Eα,β(t)1ηα,β+ηα,βetΓ(β)(t>0), (2.8)

    where

    ηα,β:=Γ(β)Γ(α+β). (2.9)

    Proof. The proof follows by applying [24, Theorem 3].

    Remark 2.6. We see that the set J is nonempty; for example, we see that (1,β)J such that β>1. For instance, (1,2)J.

    The result in the next lemma has been given in [25, Theorem 4]. We present an alternative proof.

    Lemma 2.7. For min(z,μ)>0, the following holds:

    γ(μ,z)zμeμμ+1zμ. (2.10)

    Moreover, for min(z,μ)>0, we have

    Γ(μ,z)Γ(μ)zμeμμ+1zμ. (2.11)

    Proof. Let us denote

    γ(μ,z):=γ(μ,z)zμ.

    Given (1.3), we can obtain

    γ(μ,z)=10tμ1eztdt. (2.12)

    We denote

    Fμ(z)=log(μγ(μ,z))andG(z)=z(z>0).

    We have that Fμ(0)=G(0)=0. Since the function zγ(μ,z) is log-convex on (0,) (see, for instance, the proof of Theorem 3.1 in [25]), we deduce that the function zFμ(z) is convex on (0,). This, in turn, implies that the function

    zFμ(z)=Fμ(z)G(z),

    is also increasing on (0,). Therefore, the function

    zFμ(z)G(z)=Fμ(z)Fμ(0)G(z)G(0),

    is also increasing on (0,) according to L'Hospital's rule for monotonicity [26]. Therefore, we have

    Fμ(z)G(z)limz0Fμ(z)G(z)=μμ+1.

    Then, through straightforward calculations, we can complete the proof of inequality (2.10). Finally, by combining (2.10) and (1.5), we obtain (2.11).

    The first set of main results read as follows.

    Theorem 3.1. Let b>0,z0,min(α,β)>1,b+2ν>1 and 0<2ν1. Then, the following inequalities are valid:

    πb2ν1ez(z+22b)4Eα,β(bz2)2ν12erfc(z+2b2)2Ψ(Γ)1[(ν,12,b22),(1,1)(β,α)|z]πb2ν1ez(z22b)4Eα,β(bz2)2ν12erfc(2bz2), (3.1)

    where the equality holds true if z=0: also, here, erfc is the complementary error function, defined as follows (see, e.g., [3, Eq (7.2.1)]):

    erfc(b)=2πbet2dt.

    Proof. According to Lemma 2.4, the function teatEα,β(at) is decreasing on (0,) for all min(α,β)>1 and a>0. It follows that the function tt2ν1eatEα,β(at) is decreasing on (0,) for each min(α,β)>1 and ν(0,12]. Then, for all tb, we have

    t2ν1eatEα,β(at)b2ν1eabEα,β(ab).

    Therefore

    2Ψ(Γ)1[(ν,12,b22),(1,1)(β,α)|a2]b2ν1eabEα,β(ab)2ν1bet22at2dt=b2ν1ea(a2b)2Eα,β(ab)2ν1be(ta)22dt=b2ν1ea(a2b)2Eα,β(ab)2ν1baet22dt. (3.2)

    which readily implies that the upper bound in (3.1) holds true. Now, let us focus on the lower bound of the inequalities corresponding to (3.1). We observe that the function tt2ν1et is increasing on [b,) if b+2ν1>0 and, consequently the function tt2ν1etEα,β(t) is increasing on [b,) under the given conditions. Hence,

    t2ν1eatEα,β(at)b2ν1eabEα,β(ab)(tb).

    Then, we obtain

    2Ψ(Γ)1[(ν,12,b22),(1,1)(β,α)|a2]b2ν1eabEα,β(ab)2ν1bet2+2at2dt=b2ν1ea(a+2b)2Eα,β(ab)2ν1be(t+a)22dt=b2ν1ea(a+2b)2Eα,β(ab)2ν1b+aet22dt, (3.3)

    which completes the proof of the right-hand side of the inequalities defined in (3.1). This completes the proof.

    Setting ν=13 in Theorem 3.1, we can deduce the following results.

    Corollary 3.2. For all b>13,z0, and min(α,β)>1, the following inequality holds:

    c(b)ez(z+22b)4Eα,β(bz2)erfc(z+2b2)2Ψ(Γ)1[(13,12,b22),(1,1)(β,α)|z]c(b)ez(z22b)4Eα,β(bz2)erfc(2bz2), (3.4)

    where c(b)=62π3b2.

    Example 3.3. Taking (α,β)=2 and b=12 in Corollary 3.2 gives the following statement (see Figure 1):

    L1(z):=πez(z+2)4E2,2(z2)erfc(z+12)2Ψ(Γ)1[(13,12,14),(1,1)(2,2)|z]=:ϕ1(z)πez(z2)4E2,2(z2)erfc(1z2)=:U1(z), (3.5)
    Figure 1.  The graph of the functions L1(z),ϕ1(z) and U1(z).

    Theorem 3.4. Let ν>0,min(z,b)0, and min(α,β)>1. Then,

    ez(2z+4b)42Eα,β(bz2)ϕ2ν1(z2,b)2ν12Ψ(Γ)1[(ν,12,b22),(1,1)(β,α)|z]ez(2z4b)42Eα,β(bz2)ϕ2ν1(z2,b)2ν1, (3.6)

    where ϕν(a,b) is defined by

    ϕν(a,b)=ba(t+a)νet22dt. (3.7)

    Proof. By applying part (a) of Lemma 2.4, we get

    Eα,β(at)eab+atEα,β(ab). (3.8)

    Moreover, by using the monotonicity of the function teatEα,β(at), we have

    Eα,β(at)eabatEα,β(ab). (3.9)

    Obviously, by repeating the same calculations as in Theorem 3.1, with the help of (3.8) and (3.9), we obtain (3.6).

    By applying ν=12 in (3.6), we immediately obtain the following inequalities.

    Corollary 3.5. Assume that min(z,b)0 and min(α,β)>1. Then, the following holds:

    πez(2z+4b)42Eα,β(bz2)erfc(z+2b2)2Ψ(Γ)1[(12,12,b22),(1,1)(β,α)|z]πez(2z4b)42Eα,β(bz2)erfc(2bz2), (3.10)

    where the equality holds only if z=0.

    Remark 3.6. It is worth mentioning that, if we set ν=12 in Theorem 3.1, we obtain the inequalities defined in (3.10), but under the condition b>0.

    Corollary 3.7. Under the assumptions of Corollary 3.5, the following inequalities hold:

    ez(2z+4b)42Eα,β(bz2)(e(z+2b)24πz2erfc(z+2b2))2Ψ(Γ)1[(1,12,b22),(1,1)(β,α)|z]ez(2z4b)42Eα,β(bz2)[e(z2b)24+πz2erfc(2bz2)]. (3.11)

    Proof. Taking ν=1 in (3.6) and keeping in mind the relation given by

    ϕ1(a,b)=ba(t+a)et22dt=batet22dt+abaet22dt=e(ba)22+aπ2erfc(ba2), (3.12)

    we readily establish (3.11) as well.

    Now, by making use of Corollary 3.5 and Corollary 3.7 with b=0, we obtain the following specified result.

    Corollary 3.8. For z0 and min(α,β)>1, we have

    Lα,β(z):=πez24Γ(β)erfc(z2)2Ψ1[(12,12),(1,1)(β,α)|z]=:ϕα,β(z)πez24Γ(β)erfc(z2)=:Uα,β(z). (3.13)

    By making use of Corollary 3.7 with b=0, we obtain the following specified result.

    Corollary 3.9. For z0 and min(α,β)>1, we have

    ˜Lα,β(z):=2πzez24erfc(z2)2Γ(β)2Ψ1[(1,12),(1,1)(β,α)|z]=:˜ϕα,β(z)2+πzez24erfc(z2)2Γ(β)=:˜Uα,β(z). (3.14)

    Example 3.10. If we set α=β=2 in (3.13), we obtain the following inequalities (see Figure 2):

    L2,2(z):=πez24erfc(z2)2Ψ1[(12,12),(1,1)(2,2)|z]=:ϕ2,2(z)πez24erfc(z2)=:Uα,β(z). (3.15)
    Figure 2.  The graph of the functions L2,2(z),ϕ2,2(z) and U2,2(z).

    Example 3.11. If we set α=β=2 in (3.14), we obtain the following inequalities (see Figure 3):

    ˜L2,2(z):=2πzez24erfc(z2)22Ψ1[(1,12),(1,1)(2,2)|z]=:˜ϕ2,2(z)2+πzez24erfc(z2)2=:˜U2,2(z). (3.16)
    Figure 3.  The graph of the functions ˜L2,2(z),˜ϕ2,2(z) and ˜U2,2(z).

    Theorem 3.12. Let ν>0,min(z,b)0, and (α,β)J such that α12. Then, the following inequalities hold:

    21νe(zηα,β)24Γ(β)ϕ2ν1(ηα,βz2,b)2Ψ(Γ)1[(ν,12,b22),(1,1)(β,α)|z]1ηα,βΓ(β)Γ(ν,b22)+21νηα,βez24Γ(β)ϕ2ν1(z2,b). (3.17)

    Proof. By considering the left-hand side of the inequalities defined in (2.8), i.e., where we applied the substitution u=tc(α,β), we have

    2Ψ(Γ)1[(ν,12,b22),(1,1)(β,α)|2a]21νΓ(β)bt2ν1et22aηα,βt2dt=21νe(aηα,β)22Γ(β)bt2ν1e(tηα,β)22dt=21νe(aηα,β)22Γ(β)baηα,β(u+aηα,β)2ν1eu22du, (3.18)

    which implies the right-hand side of (3.17). It remains for us to prove the left-hand side of the inequalities defined in (3.17). By applying the right-hand side of (2.8), we get

    2Ψ(Γ)1[(ν,12,b22),(1,1)(β,α)|2a]21νΓ(β)bt2ν1et22[1ηα,β+ηα,βeat]dt=21ν(1ηα,β)Γ(β)bt2ν1et22dt+21νηα,βea22Γ(β)bt2ν1e(ta)22dt=1ηα,βΓ(β)Γ(ν,b22)+21νηα,βea22Γ(β)ba(t+a)2ν1et22dt.

    Then, we can readily establish (3.17) as well.

    Corollary 3.13. For min(z,b)0 and (α,β)J such that α12, the following holds:

    πe(ηα,βz)24Γ(β)erfc(2bηα,βz2)2Ψ(Γ)1[(12,12,b22),(1,1)(β,α)|z]π(1ηα,β)Γ(β)erfc(b2)+πηα,βez24Γ(β)erfc(2bz2), (3.19)

    and the corresponding equalities hold for z=0.

    Proof. By applying ν=12 in (3.17) and performing some elementary simplifications, the asserted result described by (3.19) follows.

    As a result of b=0 in (3.19), we get the following result:

    Corollary 3.14. For ν>0 and (α,β)J such that α12, the inequalities

    πe(ηα,βz)24Γ(β)erfc(ηα,βz2)2Ψ1[(12,12),(1,1)(β,α)|z]π(1ηα,β)Γ(β)+πηα,βez24Γ(β)erfc(z2), (3.20)

    hold for all z0. Moreover, the corresponding equalities hold for z=0.

    Example 3.15. If we set ν=12,α=1, and β=2 in (3.20), we obtain the following inequalities (see Figure 4):

    L2(z):=πez216erfc(z4)2Ψ1[(12,12),(1,1)(2,1)|z]=:ϕ2(z)π2(1+ez24erfc(z2))=:U2(z), (3.21)
    Figure 4.  The graph of the functions L2(z),ϕ2(z) and U2(z).

    where z0.

    Theorem 3.16. For min(ν,z)>0 and b0, the following holds:

    2Ψ(Γ)1[(ν,12,b22),(1,1)(β,α)|z]2Ψ1[(ν,12),(1,1)(β,α)|z](b22)νeb22(2ν+b22ν)Eα,β(bz2). (3.22)

    Furthermore, if ν1,b0 and z>0, the following holds:

    2Ψ(Γ)1[(ν,12,b22),(1,1)(β,α)|z](b22)ν1eb22Eα,β(bz2). (3.23)

    Proof. By applying (2.11) we obtain

    2Ψ1[(ν,12),(1,1)(β,α)|z]2Ψ(Γ)1[(ν,12,b22),(1,1)(β,α)|z](b22)νeb22k=0eb22ν+k((zb)/2)kΓ(αk+β)(b22)νeb22k=0(1+b22ν+k)((zb)/2)kΓ(αk+β)=(b22)νeb22Eα,β(bz2)+(b22)νeb22k=0b2((zb)/2)k(2ν+k)Γ(αk+β)(b22)νeb22Eα,β(bz2)+(b22)ν+1eb22νEα,β(bz2), (3.24)

    which is equivalent to the inequality (3.22). Now, let us focus on the inequalities (3.23). By applying the following inequality [3, Eq (8.10.1)]

    Γ(μ,z)zμ1ez,(z>0,μ1). (3.25)

    Then, we get

    2Ψ(Γ)1[(ν,12,b22),(1,1)(β,α)|z](b22)ν1eb22k=0((zb)/2)kΓ(αk+β)=(b22)ν1eb22Eα,β(bz2). (3.26)

    The proof is complete.

    We recall that a real valued function f, defined on an interval I, is called completely monotonic on I if f has derivatives of all orders and satisfies

    (1)nf(n)(z)0,(nN0,zI).

    These functions play an important role in numerical analysis and probability theory. For the main properties of the completely monotonic functions, we refer the reader to [27, Chapter IV].

    Theorem 3.17. Let ν>0 and b0. If 0<α1 and βα, then the function

    z2Ψ(Γ)1[(ν,12,b22),(1,1)(β,α)|z],

    is completely monotonic on (0,). Furthermore, for 0<α1 and βα, the inequality

    2Ψ(Γ)1[(ν,12,b22),(1,1)(β,α)|z]Γ(ν,b22)exp(Γ(2ν+12,b22)Γ(α+β)Γ(ν,b22)z) (3.27)

    holds for all z>0 and b0.

    Proof. In [28], Schneider proved that the function zEα,β(z) is completely monotonic on (0,) under the parametric restrictions α(0,1] and βα (see also [29]). Then, by considering (2.1), we conclude that

    (1)kkzk(2Ψ(Γ)1[(ν,12,b22),(1,1)(β,α)|z])0(kN0,z>0).

    Finally, for inequality (3.27), we can observe that the function

    z2Ψ(Γ)1[(ν,12,b22),(1,1)(β,α)|z],

    is log-convex on (0,) since every completely monotonic function is log-convex; see [27, p. 167]. Now, for convenience, let us denote

    Φ(z):=2Ψ(Γ)1[(ν,12,b22),(1,1)(β,α)|z],F(z):=log(Φ(z)/Γ(ν,b22))andG(z)=z.

    Hence, the function zF(z) is convex on (0,) such that F(0)=0. Therefore, the function zF(z)G(z) is increasing on (0,). Again, according to L'Hospital's rule of monotonicity [26], we conclude that the function

    zF(z)G(z)=F(z)F(0)G(z)G(0),

    is increasing on (0,). Consequently,

    F(z)G(z)limz0F(z)G(z)=F(0). (3.28)

    On the other hand, by (2.1), we have

    Φ(0)=212ν2Γ(α+β)bt2νet22dt=Γ(2ν+12,b22)Γ(α+β). (3.29)

    By combining (3.28) and (3.29) via some obvious calculations, we can obtain the asserted bound (3.27).

    By setting b=0 in Theorem 3.17, we can obtain the following results:

    Corollary 3.18. Let ν>0. If 0<α1 and βα, then the function

    z2Ψ1[(ν,12),(1,1)(β,α)|z],

    is completely monotonic on (0,). Furthermore, for 0<α1 and βα, the inequality

    2Ψ1[(ν,12),(1,1)(β,α)|z]Γ(ν)exp(Γ(2ν+12)Γ(ν)Γ(α+β)z), (3.30)

    holds for all z0.

    Example 3.19. Letting ν=12,α=1, and β=2 in (3.27), we obtain the following inequality (see Figure 5):

    L3(z):=πe43πz2Ψ1[(12,12),(1,1)(2,1)|z]=:ϕ3(z),z>0. (3.31)
    Figure 5.  The graph of the functions L3(z) and ϕ3(z).

    Remark 3.20. As in Section 3, we may derive new upper and/or lower bounds for the lower incomplete Fox-Wright function 2Ψ(γ)1[z], by simple replacing the relation (2.1) with the following relation:

    2Ψ(γ)1[(ν,12,b22),(1,1)(β,α)|z]=21νb0t2ν1et22Eα,β(zt2)dt, (3.32)
    (min(z,ν,β)>0,b0,α12).

    In [6, Section 6], Srivastava et al. presented several applications for the incomplete Fox-Wright functions in communication theory and probability theory. It is believed that certain forms of the incomplete Fox-Wright functions, which we have studied here, have the potential for application in fields similar to those mentioned above, including probability theory.

    In our present investigation, we have established new functional bounds for a class of functions that are related to the lower incomplete Fox-Wright functions; see (1.7). We have also presented a class of completely monotonic functions related to the aforementioned type of function. In particular, we have reported on bilateral functional bounds for the Fox-Wright function 2Ψ1[.]. Moreover, we have presented some conditions to be imposed on the parameters of the Fox-Wright function 2Ψ1[.], and these conditions have allowed us to conclude that the function is completely monotonic. Some applications of this type of incomplete special function have been discussed for probability theory.

    The mathematical tools that have been applied in the proofs of the main results in this paper will inspire and encourage the researchers to study new research directions that involve the formulation of some other special functions related to the incomplete Fox-Wright functions, such as the Nuttall Q-function [14], the generalized Marcum Q-function, and Marcum Q-function. Yet another novel direction of research can be pursued for other special functions when we replace the two-parameter Mittag-Leffler function with other special functions such as the three-parameter Mittag-Leffler function (or Prabhakar's function [30]), the two-parameter Wright function [2], and the four parameter Wright function; see [31, Eq (21)].

    Khaled Mehrez and Abdulaziz Alenazi: Writing–original draft; Writing–review & editing. All authors have read and agreed to the published version of the manuscript.

    The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

    The authors extend their appreciation to the Deanship of Scientific Research at Northern Border University, Arar, KSA for funding this research work through the project number "NBU-FPEJ-2024-220-01".

    The authors declare that they have no conflicts of interest.



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