Loading [MathJax]/jax/output/SVG/jax.js
Mini review

Acute infection with measles virus predisposes to mastoiditis with concomitant facial paralysis and neck abscess: A minireview of pathomechanism and diagnostic approach

  • Received: 25 August 2020 Accepted: 12 October 2020 Published: 20 October 2020
  • Despite the availability of safe, reliable, and cost-effective measles vaccine, we continue to experience dreadful measles outbreaks with devastating multisystem complications, especially in the pediatric age group. In most instances, the complications arise from a late presentation or delayed institution of appropriate care. With co-existence of measles virus and bacteria in the middle ear, suppurative otitis media can involve the mastoid process and causes fatal complications that manifest late when the patient is in a dire state. This short review highlights the pathogenic mechanisms leading to mastoiditis, facial paralysis, and neck abscess following acute infection with the measles virus, and outlines some useful diagnostic tips. In this review, we searched the international electronic database (PubMed, Web of Science, and Embase) and Google Scholar for articles published on complications of acute measles infection. The keywords used were “mastoiditis”, “mastoid antrum”, “middle ear”, “otitis media”, “Bezold's abscess”, “facial paralysis” with an operator “OR”; “AND” measles; with restriction to the English language. Also, we searched for similar information in the local clinical and virology journals databases. Thereafter, we reviewed the publications and we described the findings qualitatively.

    Citation: Auwal Idris Kabuga, Yusuf Ashiru Hassan, Muhammad Ibrahim Getso. Acute infection with measles virus predisposes to mastoiditis with concomitant facial paralysis and neck abscess: A minireview of pathomechanism and diagnostic approach[J]. AIMS Medical Science, 2020, 7(4): 269-277. doi: 10.3934/medsci.2020016

    Related Papers:

    [1] Hari Mohan Srivastava, Muhammad Arif, Mohsan Raza . Convolution properties of meromorphically harmonic functions defined by a generalized convolution $ q $-derivative operator. AIMS Mathematics, 2021, 6(6): 5869-5885. doi: 10.3934/math.2021347
    [2] Caihuan Zhang, Shahid Khan, Aftab Hussain, Nazar Khan, Saqib Hussain, Nasir Khan . Applications of $ q $-difference symmetric operator in harmonic univalent functions. AIMS Mathematics, 2022, 7(1): 667-680. doi: 10.3934/math.2022042
    [3] Bakhtiar Ahmad, Muhammad Ghaffar Khan, Basem Aref Frasin, Mohamed Kamal Aouf, Thabet Abdeljawad, Wali Khan Mashwani, Muhammad Arif . On $ q $-analogue of meromorphic multivalent functions in lemniscate of Bernoulli domain. AIMS Mathematics, 2021, 6(4): 3037-3052. doi: 10.3934/math.2021185
    [4] Alina Alb Lupaş, Shujaat Ali Shah, Loredana Florentina Iambor . Fuzzy differential subordination and superordination results for $ q $ -analogue of multiplier transformation. AIMS Mathematics, 2023, 8(7): 15569-15584. doi: 10.3934/math.2023794
    [5] Murugusundaramoorthy Gangadharan, Vijaya Kaliyappan, Hijaz Ahmad, K. H. Mahmoud, E. M. Khalil . Mapping properties of Janowski-type harmonic functions involving Mittag-Leffler function. AIMS Mathematics, 2021, 6(12): 13235-13246. doi: 10.3934/math.2021765
    [6] Shujaat Ali Shah, Ekram Elsayed Ali, Adriana Cătaș, Abeer M. Albalahi . On fuzzy differential subordination associated with $ q $-difference operator. AIMS Mathematics, 2023, 8(3): 6642-6650. doi: 10.3934/math.2023336
    [7] Nusrat Raza, Mohammed Fadel, Kottakkaran Sooppy Nisar, M. Zakarya . On 2-variable $ q $-Hermite polynomials. AIMS Mathematics, 2021, 6(8): 8705-8727. doi: 10.3934/math.2021506
    [8] Ahmad A. Abubaker, Khaled Matarneh, Mohammad Faisal Khan, Suha B. Al-Shaikh, Mustafa Kamal . Study of quantum calculus for a new subclass of $ q $-starlike bi-univalent functions connected with vertical strip domain. AIMS Mathematics, 2024, 9(5): 11789-11804. doi: 10.3934/math.2024577
    [9] Ekram E. Ali, Miguel Vivas-Cortez, Rabha M. El-Ashwah . New results about fuzzy $ \mathbf{\gamma } $-convex functions connected with the $ \mathfrak{q} $-analogue multiplier-Noor integral operator. AIMS Mathematics, 2024, 9(3): 5451-5465. doi: 10.3934/math.2024263
    [10] Ekram E. Ali, Georgia Irina Oros, Rabha M. El-Ashwah, Abeer M. Albalahi . Applications of fuzzy differential subordination theory on analytic $ p $ -valent functions connected with $ \mathfrak{q} $-calculus operator. AIMS Mathematics, 2024, 9(8): 21239-21254. doi: 10.3934/math.20241031
  • Despite the availability of safe, reliable, and cost-effective measles vaccine, we continue to experience dreadful measles outbreaks with devastating multisystem complications, especially in the pediatric age group. In most instances, the complications arise from a late presentation or delayed institution of appropriate care. With co-existence of measles virus and bacteria in the middle ear, suppurative otitis media can involve the mastoid process and causes fatal complications that manifest late when the patient is in a dire state. This short review highlights the pathogenic mechanisms leading to mastoiditis, facial paralysis, and neck abscess following acute infection with the measles virus, and outlines some useful diagnostic tips. In this review, we searched the international electronic database (PubMed, Web of Science, and Embase) and Google Scholar for articles published on complications of acute measles infection. The keywords used were “mastoiditis”, “mastoid antrum”, “middle ear”, “otitis media”, “Bezold's abscess”, “facial paralysis” with an operator “OR”; “AND” measles; with restriction to the English language. Also, we searched for similar information in the local clinical and virology journals databases. Thereafter, we reviewed the publications and we described the findings qualitatively.


    An analytic function s:U={z:|z|<1}C is subordinate to an analytic function t:UC and write s(z)t(z), if there exists a complex value function ω which maps U into itself with ω(0)=0and|ω(z)|<1(zU) such that s(z)=t(ω(z))(zU). Furthermore, if the function t is univalent in U, then we have the following equivalence (see [1]):

    s(z)t(z)s(0)=t(0)ands(U)t(U).

    Let A define the class of functions f that are analytic in the open unit disc U of the form

    f(z)=z+k=2akzk.

    The theory of (p,q)-calculus (or post quantum calculus [2]) operators are used in various areas of science and also in geometric function theory. For 0<qp1 and fA, Chakrabarti and Jagannathan [2] defined the (p,q)-derivative operator Dp,q:AA by

    Dp,qf(z)={f(pz)f(qz)(pq)z,pq,z0,limqpf(pz)f(qz)(pq)z,p=q,z0, (1.1)

    where

    Dp,qf(z)=1+k=2[k]p,qakzk1 (1.2)

    and

    [k]p,q=pkqkpq={k=1p1q(k1)(1),pq,kpk1,p=q. (1.3)

    From (1.1), we have

    limz0Dp,qf(z)=1  and  limp1Dp,pf(z)=D1,1f(z)=f(z).

    Next, we introduce the (p,q)-derivative operator in the class of meromorphic functions.

    Suppose M be the class of functions f that are meromorphic analytic in the punctured disk U=U{0}={z:0<|z|<1} of the form

    f(z)=1z+k=1akzk. (1.4)

    Now, we define the (p,q)-post quantum derivative operator ˜dp,q:MM by

    ˜dp,qf(z)={f(pz)f(qz)(pq)z,pq,zU,limqpf(pz)f(qz)(pq)z,p=q,zU. (1.5)

    Using (1.4) and (1.5), we have

    ˜dp,qf(z)=1pqz2+k=1[k]p,qakzk1  (kN), (1.6)

    where 0<qp1 and [k]p,q is defined by (1.3).

    Let λ0,0<qp1,mN0=N{0} and f(z)M, we introduce the generalized (p,q)-post quantum calculus operator ˜Dm,λp,q:MM as follows,

    ˜D0,0p,qf(z)=f(z),˜D1,λp,qf(z)=(1λ)pqz˜dp,qf(z)+λpqz(z˜dp,qf(z)))+2(λ+1)z=˜Dλp,qf(z), (1.7)
    ˜D2,λp,qf(z)=˜Dλp,q(˜Dλp,qf(z)) (1.8)

    and in general,

    ˜Dm,λp,qf(z)=˜Dλp,q(˜Dm1,λp,qf(z))(m1,zU). (1.9)

    After a simple calculation, we can obtain the following conclusion

    ˜Dm,λp,qf(z)=1z+k=1{[(k1)λ+1]pq[k]p,q}makzk, (1.10)

    where [k]p,q is defined by (1.3). For simple of notation, we let

    ωk(λ,p,q):=[(k1)λ+1]pq[k]p,q. (1.11)

    Obviously, for λ=0, the operator ˜Dm,0p,qf(z)=Lmp,qf(z) reduces to the (p,q)-Sǎlǎgean operator [3].

    A complex valued harmonic function f in a simply connected domain DC has the canonical representation f=h+¯g, where h and g are analytic in D and g(z0)=0 for some prescribed point z0D. A necessary and sufficient condition for f to be locally univalent and sense preserving in D is that |h(z)|>|g(z)| in D (see [4,5]).

    Denote by MH the class of meromorphic univalent and harmonic functions f that are sense preserving in U and have the following form

    f(z)=h(z)+¯g(z)=1z+k=1akzk+k=1¯bkzk,|b1|<1, (1.12)

    where h(z) and g(z) are analytic in U and U respectively. The class MH was studied in [6,7,8,9,10].

    Let λ0,0<qp1,mN0 and fMH, we now define the operator ˜Dm,λp,q:MHMH as

    ˜Dm,λp,qf(z)=˜Dm,λp,qh(z)+¯˜Dm,λp,qg(z), (1.13)

    where

    ˜Dm,λp,qh(z)=1z+k=1ωk(λ,p,q)akzk,˜Dm,λp,qg(z)=k=1ωk(λ,p,q)bkzk, (1.14)

    with ωk(λ,p,q) defined by (1.11).

    Assume that F be fixed meromorphic harmonic function given by

    F(z)=H(z)+¯G(z)=1z+k=1Akzk+¯k=1Bkzk,|B1|<1. (1.15)

    For f given by (1.12) and F given by (1.15), we define the convolution (or Hadamard product) of F and f by

    (fF)(z):=1z+k=1akAkzk+¯k=1bkBkzk=(Ff)(z). (1.16)

    Also, we denote by T(TMH) the class of meromorphic harmonic functions f of the following form

    f(z)=h(z)+¯g(z)=1z+k=1|ak|zkk=1|bk|¯zk  (zU). (1.17)

    Throughout this paper, we shall assume λ0,0<qp1,mN0 and 1B<A1.

    Let

    ϕ(z)=1z+k=1ukzk+k=1¯vkzk (1.18)

    be harmonic in U with uk>0 and vk>0.

    Taking

    LHf(z)=zh(z)¯zg(z),L2Hf(z)=LH(LHf(z)),fMH.

    Now, using the operator ˜Dm,λp,q and subordination relationship, we define the following two classes.

    Definition 1. Let the function fMH of the form (1.12). The function fMp,qϕ(λ,m,A,B) if and only if

    LH(˜Dm,λp,qfϕ)(z)(˜Dm,λp,qfϕ)(z)1+Az1+Bz (1.19)

    and also the function fKp,qϕ(λ,m,A,B) if and only if

    L2H(˜Dm,λp,qfϕ)(z)LH(˜Dm,λp,qfϕ)(z)1+Az1+Bz, (1.20)

    where

    ˜Dm,λp,q(fϕ)(z)=1z+k=1ωmk(λ;p,q)ukakzk+k=1ωmk(λ;p,q)vk¯bkzk (1.21)

    with ωk(λ;p,q) given by (1.11).

    We let

    ˜Mp,qϕ(λ,m,A,B)=TMp,qϕ(λ,m,A,B)

    and

    ˜Kp,qϕ(λ,m,A,B)=TKp,qϕ(λ,m,A,B).

    The classes Mp,qϕ(λ,m,A,B) and Kp,qϕ(λ,m,A,B) reduce to the well-known subclasses of MH as well as many new ones. For example, let ϕ(z)=1z+k=1(zk+ˉzk), we have

    M1,1ϕ(0,1,12γ,1)=MHS(γ)={fMH:Re[zh(z)¯zg(z)h(z)+¯g(z)]>γ}

    and

    K1,1ϕ(0,1,12γ,1)=MCH(γ)={fMH:Re[zh(z)+h(z)+¯zg(z)+g(z)h(z)¯g(z)]>γ},

    where γ[0,1).

    The classes MHS(γ) and MCH(γ) were studied by Jahangiri [9].

    In particular, the classes MHS(0)=MHS (Meromorphically harmonic starlike functions) and MCH(0)=MCH (Meromorphically harmonic convex functions) were studied by Jahangiri and Silverman [10].

    In this paper, the sufficient and necessary conditions of coefficients are discussed. As what we have hoped, distortion estimates, extreme points and convolution properties for the above-defined classes are also obtained.

    First of all, we provide the sufficient conditions of coefficients for the classes defined in Definition 1.

    Theorem 1. Let f=h+¯g be given by (1.12) and ωk(λ;p,q) given by (1.11).

    (i) The sufficient condition for f to be sense-preserving and meromorphic harmonic univalent in U and fMp,qϕ(λ,m,A,B) is

    k=1[ξk,m(p,q)|ak|+μk,m(p,q)|bk|]1, (2.1)

    where

    {kξk,m(p,q):=[k(1B)+(1A)]ukωmk(λ;p,q)AB,kμk,m(p,q):=[k(1B)(1A)]vkωmk(λ;p,q)AB. (2.2)

    (ii) The sufficient condition for f to be sense-preserving and meromorphic harmonic univalent in U and fKp,qϕ(λ,m,A,B) is

    k=1k[ξk,m(p,q)|ak|+μk,m(p,q)|bk|]1, (2.3)

    where ξk,m(p,q) and μk,m(p,q) are given by (2.2).

    Proof. (i) For 0<|z1||z2|<1, we obtain

    |f(z1)f(z2)h(z1)h(z2)|1|g(z1)g(z2)h(z1)h(z2)|=1|z1z2k=1bk(zk1zk2)(z1z2)z1z2k=1ak(zk1zk2)|>1k=1k|bk|1k=1k|ak|1k=1μk,m(p,q)|bk|1k=1ξk,m(p,q)|ak|0,

    which proves univalence. Note that f is sense-preserving harmonic in U. This is because

    |h(z)|1|z|2k=1k|ak||z|k1>1k=1ξk,m(p,q)|ak|k=1μk,m(p,q)|bk|>k=1k|bk||z|k1|g(z)|.

    Next, we show that if the inequality (2.1) holds, then the required condition (1.19) is satisfied.

    By means of Definition 1 and relationship of subordination, the function fMp,qϕ(λ,m,A,B) iff there exists an analytic function ϖ(z) satisfying ϖ(0)=0,|ϖ(z)|<1(zU) such that

    LH(Dm,λp,qfϕ)(z)Dm,λp,qfϕ(z)=1+Aϖ(z)1+Bϖ(z),

    or equivalently

    |LH(Dm,λp,qfϕ)(z)+Dm,λp,qfϕ(z)ADm,λp,qfϕ(z)+BLH(Dm,λp,qfϕ)(z)|<1.

    We only need to show that

    |ADm,λp,qfϕ(z)+BLH(Dm,λp,qfϕ)(z)||LH(Dm,λp,qfϕ)(z)+Dm,λp,qfϕ(z)|>0(zU). (2.4)

    Letting

    {σk,j=(A+(1)j1kB)ωmk(λ;p,q),j=1,2,θk,j=(k+(1)j1)ωmk(λ;p,q),j=1,2. (2.5)

    Therefore, from (2.1) we get

    |ADm,λp,qfϕ(z)+BLH(Dm,λp,qfϕ)(z)||LH(Dm,λp,qfϕ)(z)+Dm,λp,qfϕ(z)|=|(AB)1z+k=1σk,1ukakzk+k=1σk,2vk¯bkzk||k=1θk,1ukakzk+k=1θk,2vk¯bkzk|(AB)1|z|+k=1σk,1uk|ak||z|kk=1σk,2vk|bk||z|kk=1θk,1uk|ak||z|kk=1θk,2vk|bk||z|k=(AB)1|z|[1k=1ξk,m(p,q)|ak||z|k+1k=1μk,m(p,q)|bk||z|k+1]>(AB)1|z|[1k=1ξk,m(p,q)|ak|k=1μk,m(p,q)|bk|]0.

    Hence, we complete the proof of (i). Also, applying the same method as (i), we can obtain (ii).

    The harmonic univalent function

    f(z)=1z+k=1ABukωmk(λ,p,q)[k(1B)+(1A)]xkzk+ABvkωmk(λ,p,q)[k(1B)(1A)]¯ykzk, (2.6)

    where k=1(|xk|+|yk|)=1, shows that the coefficient bound given by (2.1) is sharp.

    Theorem 2. Let f=h+¯g be given by (1.17). Then

    (i) f˜Mp,qϕ(λ,m,A,B) iff (2.1) holds true.

    (ii) f˜Kp,qϕ(λ,m,A,B) iff (2.3) holds true.

    Proof. (i) It appears from (1.17) that ˜Mp,qϕ(λ,m,A,B)Mp,qϕ(λ,m,A,B). In view of Theorem 1, it is straightforward to show that if f˜Mp,qϕ(λ,m,A,B), then (2.1) holds true. Next, we use the method in [11] to prove.

    Let f˜Mp,qϕ(λ,m,A,B), then it satisfies (1.19) or equivalently

    |k=1θk,1uk|ak|zk+k=1θk,2vk|bk|ˉzk(AB)1z+k=1σk,1uk|ak|zkk=1σk,2vk|bk|ˉzk|<1(zU). (2.7)

    From (2.7), we get

    Re{k=1θk,1uk|ak|zk+k=1θk,2vk|bk|ˉzk(AB)1z+k=1σk,1uk|ak|zkk=1σk,2vk|bk|ˉzk}<1, (2.8)

    which holds for all zU. Setting z=r(0<r<1) in (2.8), we get

    k=1θk,1uk|ak|rk+1+k=1θk,2vk|bk|rk+1(AB)+k=1σk,1uk|ak|rk+1k=1σk,2vk|bk|rk+1<1. (2.9)

    Thus, from (2.9) we have

    k=1[ξk,m(p,q)|ak|+μk,m(p,q)|bk|]rk+1<1(0<r<1), (2.10)

    where ξk,m(p,q) and μk,m(p,q) are given by (2.2).

    Putting

    Sn=nk=1(ξk,m(p,q)|ak|+μk,m(p,q)|bk|).

    For the series k=1[ξk,m(p,q)|ak|+μk,m(p,q)|bk|], {Sn} is the nondecreasing sequence of partial sums of it. Moreover, by (2.10) it is bounded by 1. Therefore, it is convergent and

    k=1(ξk,m(p,q)|ak|+μk,m(p,q)|bk|)=limnSn1.

    Thus, we get the inequality (2.1). Similarly, it is easy to prove (ii) of Theorem 2.

    Clearly, from Theorem 2, we have

    ˜Kp,qϕ(λ,m,A,B)˜Mp,qϕ(λ,m,A,B). (2.11)

    Next, we give the extreme points of these classes.

    Theorem 3. Let Xk0,Yk0,k=0Xk+k=1Yk=1,ξk,m(p,q) and μk,m(p,q) be given by (2.2).

    (i) If f˜Mp,qϕ(λ,m,A,B). Then fclco˜Mp,qϕ(λ,m,A,B) iff

    f(z)=k=0Xkhk+k=1Ykgk(zU), (2.12)

    where

    {h0=1z,hk=1z+1ξk,m(p,q)zk,k1,gk=1z1μk,m(p,q)ˉzk,k1. (2.13)

    (ii) If f˜Kp,qϕ(λ,m,A,B). Then fclco˜Kp,qϕ(λ,m,A,B) iff the condition (2.12) holds and

    {h0=1z,hk=1z+1kξk,m(p,q)zk,k1,gk=1z1kμk,m(p,q)ˉzk,k1. (2.14)

    Proof. From (2.12) we get

    f(z)=(X0+k=1[Xk+Yk])1z+k=11ξk,m(p,q)Xkzkk=11μk,m(p,q)Yk¯zk.

    Since 0Xk1(k=0,1,2,), we obtain

    k=1ξk,m(p,q)1ξk,m(p,q)Xk+k=1μk,m(p,q)1μk,m(p,q)Yk=k=1Xk+Yk=1X01.

    It follows, from (i) of Theorem 2, that f˜Mp,qϕ(λ,m,A,B).

    Conversely, if f˜Mp,qϕ(λ,m,A,B), then

    |ak|1ξk,m(p,q)  and  |bk|1μk,m(p,q).

    Putting Xk=ξk,m(p,q)|ak|,Yk=μk,m(p,q)|bk| and X0=1k=1Xkk=1Yk0, we obtain

    f(z)=1z+k=1|ak|zkk=1|bk|¯zk=(k=0Xk+k=1Yk)1z+k=11ξk,m(p,q)Xkzkk=11μk,m(p,q)Yk¯zk=k=0hk(z)Xk+k=1gk(z)Yk.

    Thus f can be expressed in the form of (2.12). The remainder of the proof is analogous to (i) in Theorem 3 and so we omit.

    Next, using Theorem 2, we proceed to discuss the distortion theorems for functions of these classes.

    Theorem 4. Let f=h+¯g be of the form (1.17), |z|=r(0,1), ξk,m(p,q) and μk,m(p,q) are defined by (2.2), {ξk,m(p,q)} and {μk,m(p,q)} are non-decreasing sequences. If f˜Mp,qϕ(λ,m,A,B), then

    1rrmin{ξ1,m(p,q),μ1,m(p,q)}|f(z)|1r+rmin{ξ1,m(p,q),μ1,m(p,q)}.

    Proof. For f˜Mp,qϕ(λ,m,A,B), using Theorem 2 and (2.1), we have

    |f(z)|=|1z+k=1|ak|zkk=1|bk|¯zk|1r+1min{ξ1,m(p,q),μ1,m(p,q)}k=1(ξk,m(p,q)|ak|+μk,m(p,q)|bk|)r1r+1min{ξ1,m(p,q),μ1,m(p,q)}r

    and

    |f(z)|1r(k=1|ak|+k=1|bk|)r1r1min{ξ1,m(p,q),μ1,m(p,q)}r.

    The result is sharp and the extremal function is

    f(z)=1z1min{ξ1,m(p,q),μ1,m(p,q)}z.

    So, we complete the proof of Theorem 4.

    By virtue of Theorem 4, we obtain the following covering result.

    Theorem 5. Let ξk,m(p,q) and μk,m(p,q) be given by (2.2). If f˜Mp,qϕ(λ,m,A,B), then

    {w:|w|<11min{ξ1,m(p,q),μ1,m(p,q)}}f(U).

    Theorem 6. The classes ˜Mp,qϕ(λ,m,A,B) and ˜Kp,qϕ(λ,m,A,B) are closed under convex combinations.

    Remark 1. By taking the special values of the parameters λ,p,q,m,A,B and ϕ in Theorems 1-6, it is easy to show the corresponding results for the classes MHS(γ) and MCH(γ) which are defined in Section 1.

    Especially, let λ=0,p=q=m=1,A=12γ,B=1,0γ<1 and ϕ(z)=1z+k=1(zk+ˉzk) in Theorem 2, we can obtain the results of Theorems 1 and 7 in [9].

    Corollary 1. Let f=h+¯g be given by (1.17). Then

    (i) fMHS(γ) iff

    k=1k+γ1γ|ak|+kγ1γ|bk|1.

    (ii) fMCH(γ) iff

    k=1k(k+γ)1γ|ak|+k(kγ)1γ|bk|1.

    Next, in order to obtain the convolution properties of functions belonging to the classes ˜Mp,qϕ(λ,m,A,B) and ˜Kp,qϕ(λ,m,A,B), we now introduce a new class of harmonic functions.

    Definition 2. Let δ0, the function f=h+¯g of the form (1.17) belongs to the class ˜Lδ,p,qϕ(λ,m,A,B) if and only if

    k=1kδξk,m(p,q)|ak|+k=1kδμk,m(p,q)|bk|1, (3.1)

    where ξk,m(p,q) and μk,m(p,q) are defined by (2.2).

    Obviously, for any positive integer δ, we have the following inclusion relation:

    ˜Lδ,p,qϕ(λ,m,A,B)˜Kp,qϕ(λ,m,A,B)˜Mp,qϕ(λ,m,A,B). (3.2)

    Let the harmonic functions ft(t=1,2,,ρ) and Fl(l=1,2,,η) of the following form

    ft(z)=ht(z)+¯gt(z)=1z+k=1|ak,t|zkk=1|bk,t|¯zk,|b1,t|<1 (3.3)

    and

    Fl(z)=Hl(z)+¯Gl(z)=1z+k=1|Ak,l|zkk=1|Bk,l|¯zk,|B1,|<1. (3.4)

    We define the Hadamard product (or convolution) of ft and F by

    (ftFl)(z):=1z+k=1|ak,t||Ak,l|zkk=1|bk,t||Bk,l|¯zk=:(Flft)(z), (3.5)

    where t=1,2,,ρ and l=1,2,,η.

    Using Theorem 2, we obtain the following results.

    Theorem 7. Let ft of the form (3.3) be in the class ˜Kp,qϕ(λ,m,A,B)(t=1,2,,ρ) and Fl of the form (3.4) be in the class ˜Mp,qϕ(λ,m,A,B)(l=1,2,,η). Then the Hadamard product (f1f2fρF1F2Fη)(z) belongs to the class ˜Lδ,p,qϕ(λ,m,A,B), where δ=2ρ+η1.

    Proof. Using the method in [8] to prove the theorem. Putting

    χ(z)=(f1f2fρF1F2Fη)(z). (3.6)

    From (3.6) we have

    χ(z)=1z+k=1(ρt=1|ak,t|ηl=1|Ak,l|)zkk=1(ρt=1|bk,t|ηl=1|Bk,l|)¯zk. (3.7)

    According to Definition 2, we only need to show that

    k=1k2ρ+η1ξk,m(p,q)(ρt=1|ak,t|ηl=1|Ak,l|)+k=1k2ρ+η1μk(p,q)(ρt=1|bk,t|ηl=1|Bk,l|)1, (3.8)

    where ξk,m(p,q) and μk,m(p,q) are defined by (2.2).

    For ft˜Kp,qϕ(λ,m,A,B), we obtain

    k=1kξk,m(p,q)|ak,t|+k=1kμk,m(p,q)|bk,t|1, (3.9)

    for every t=1,2,,ρ. Therefore

    kξk,m(p,q)|ak,t|1  and  kμk,m(p,q)|bk,t|1. (3.10)

    Further, by ξk,m(p,q)k and μk,m(p,q)k, we have

    |ak,t|k2and|bk,t|k2  (t=1,2,,ρ). (3.11)

    Also, since Fl˜Mp,qϕ(λ,m,A,B), we have

    k=1ξk,m(p,q)|Ak,l|+k=1μk,m(p,q)|Bk,l|1  (l=1,2,,η). (3.12)

    Hence we obtain

    |Ak,l|k1and|Bk,l|k1  (l=1,2,,η). (3.13)

    Using (3.11) for t=1,2,,ρ, (3.13) for l=1,2,,η1 and (3.12) for l=η, we obtain

    k=1k2ρ+η1ξk,m(p,q)(ρt=1|ak,t|η1l=1|Ak,l|)|Ak,η|+k=1k2ρ+η1μk,m(p,q)(ρt=1|bk,t|η1l=1|Bk,l|)|Bk,η|k=2k2ρ+η1(ξk,m(p,q)k2ρk(η1))|Ak,η|+k=1k2ρ+η1(μk,m(p,q)k2ρk(η1))|Bk,η|=k=1ξk,m(p,q)|Ak,l|+k=1μk,m(p,q)|Bk,l|1,

    and therefore χ(z)˜Lδ,p,qϕ(λ,m,A,B),δ=2ρ+η1. We note that the required estimate can also be obtained by using (3.11) for t=1,2,,η1; (3.13) for l=1,2,,η and (3.9) for t=ρ.

    Taking into account the Hadamard product of functions f1f2fρ only, in the proof of Theorem 3.3, and using (3.11) for t=1,2,,ρ1; and relation (3.9) for t=ρ, we are led to

    Corollary 2. Let the functions ft defined by (3.3) be in the class ˜Kp,qϕ(λ,m,A,B) for every t=1,2,,ρ. Then the Hadamard product (f1f2fρ)(z) belongs to the class ˜L2ρ1,p,qϕ(λ,m,A,B).

    Also, taking into account the Hadamard product of functions F1F2Fη only, in the proof of Theorem 3.3, and using (3.13) for l=1,2,,η1; and relation (3.12) for l=η, we are led to

    Corollary 3. Let the functions Fm,l defined by (3.4) be in the class ˜Mp,qϕ(λ,m,A,B) for every l=1,2,,η. Then the Hadamard product (F1F2Fη)(z) belongs to the class ˜Lη1,p,qϕ(λ,m,A,B).

    Remark 2. For different choices of the parameters λ,p,q,m,A,B and ϕ in Theorem 7, we can deduce some new results for each of the following univalent harmonic function classes MHS(γ) and MCH(γ) which are defined in Section 1.

    This work was supported by Natural Science Foundation of Inner Mongolia Autonomous Region of China (Grant No. 2019MS01023; Grant No. 2020MS01011; Grant No. 2018MS01026) and Research Program of science and technology at Universities of Inner Mongolia Autonomous Region (Grant No. NJZZ19209; Grant No. NJZY20198).

    The authors agree with the contents of the manuscript, and there is no conflict of interest among the authors.


    Acknowledgments



    We acknowledge Dr. Majid Teymoor-Rad, Department of Virology, School of Public Health, Tehran University of Medical Sciences, Tehran for drawing the anatomic description used in this article. We are also grateful to authors/editors/publishers of all the articles, journals, and books from where the literature for this article has been reviewed and discussed.

    Conflict of interest



    There was no financial support from any institution to this research. The authors declare that they have no conflict of interest.

    [1] Sansonetti PJ (2018) Measles 2018: A tale of two anniversaries. EMBO Mol Med 10: e9176. doi: 10.15252/emmm.201809176
    [2] Onoja A, Adeniji A, Faneye A (2013) Measles complications in a Nigerian hospital setting. Clin Rev Opinions 5: 18-23. doi: 10.5897/CRO12.008
    [3] Duru CO, Peterside O, Adeyemi OO (2014) A 5 year review of childhood measles at the Niger Delta University Teaching Hospital, Bayelsa state, Nigeria. J Med Med Sci 5: 78-86.
    [4] Zipprich J, Winter K, Hacker J, et al. (2015) Measles outbreak—California, December 2014–February 2015. MMWR Morb Mortal Wkly Rep 64: 153-154.
    [5] Wendorf KA, Winter K, Zipprich J, et al. (2017) Subacute sclerosing panencephalitis: The devastating measles complication that might be more common than previously estimated. Clin Infect Dis 65: 226-232. doi: 10.1093/cid/cix302
    [6] Weldegebriel GG, Gasasira A, Harvey P, et al. (2011) Measles resurgence following a nationwide measles vaccination campaign in Nigeria, 2005–2008. J Infect Dis 204: S226-S231. doi: 10.1093/infdis/jir136
    [7] Singh SK (2014)  Human respiratory viral infections CRC Press. doi: 10.1201/b16778
    [8] Suboti R (1976) Histopathological findings in the inner ear caused by measles. J Laryngol Otol 90: 173-181. doi: 10.1017/S0022215100081937
    [9] Mustafa A, Toçi B, Thaçi H, et al. (2018) Acute mastoiditis complicated with concomitant Bezold's abscess and lateral sinus thrombosis. Case Rep Otolaryngol 2018.
    [10] Leibovitz E (2008) Complicated otitis media and its implications. Vaccine 26: G16-G19. doi: 10.1016/j.vaccine.2008.11.008
    [11] Tatsuo H, Ono N, Tanaka K, et al. (2000) SLAM (CDw150) is a cellular receptor for measles virus. Nature 406: 893-897. doi: 10.1038/35022579
    [12] De Witte L, De Vries RD, Van Der Vlist M, et al. (2008) DC-SIGN and CD150 have distinct roles in transmission of measles virus from dendritic cells to T-lymphocytes. PLoS Pathog 4: e1000049. doi: 10.1371/journal.ppat.1000049
    [13] Buckland R, Wild TF (1997) Is CD46 the cellular receptor for measles virus? Virus Res 48: 1-9. doi: 10.1016/S0168-1702(96)01421-9
    [14] Noyce RS, Bondre DG, Ha MN, et al. (2011) Tumor cell marker PVRL4 (nectin 4) is an epithelial cell receptor for measles virus. PLoS Pathog 7: e1002240. doi: 10.1371/journal.ppat.1002240
    [15] Laksono BM, de Vries RD, Duprex WP, et al. (2020) Measles pathogenesis, immune suppression and animal models. Curr Opin Virol 41: 31-37. doi: 10.1016/j.coviro.2020.03.002
    [16] Laksono BM, Grosserichter-Wagener C, de Vries RD, et al. (2018) In vitro measles virus infection of human lymphocyte subsets demonstrates high susceptibility and permissiveness of both naive and memory B cells. J Virol 92: e00131-18. doi: 10.1128/JVI.00131-18
    [17] Laksono BM, de Vries RD, Verburgh RJ, et al. (2018) Studies into the mechanism of measles-associated immune suppression during a measles outbreak in the Netherlands. Nat Commun 9: 1-10. doi: 10.1038/s41467-018-07515-0
    [18] Laksono BM, de Vries RD, McQuaid S, et al. (2016) Measles virus host ınvasion and pathogenesis. Viruses 8: 210. doi: 10.3390/v8080210
    [19] Perry RT, Halsey NA (2004) The clinical significance of measles: A review. J Infect Dis 189: S4-S16. doi: 10.1086/377712
    [20] Kurle G (2017) Bezold's abscess: A rare complication of chronic suppurative otitis media in a 13 year old female child. Int J Otorhinolaryngol Head Neck Surg 3: 437. doi: 10.18203/issn.2454-5929.ijohns20171194
    [21] Oestreicher-Kedem Y, Popovtzer A, Raveh E, et al. (2005) Complications of mastoiditis in children at the onset of a new millennium. Ann Otol Rhinol Laryngol 114: 147-152. doi: 10.1177/000348940511400212
    [22] Antoli-Candela F, Stewart T (1974) The pathophysiology of otologic facial paralysis. Otolaryngol Clin North Am 7: 309-330. doi: 10.1016/S0030-6665(20)32839-5
    [23] Moreano EH, Paparella MM, Zelterman D, et al. (1994) Prevalence of facial canal dehiscence and of persistent stapedial artery in the human middle ear: A report of 1000 temporal bones. Laryngoscope 104: 309-320.
    [24] Selesnick SH, Jackler RK (1992) Facial paralysis in suppurative ear disease. Oper Tech Otolaryngol Head Neck Surg 3: 61-68. doi: 10.1016/S1043-1810(10)80014-4
    [25] Elliott CA, Zalzal GH, Gottlieb WR (1996) Acute otitis media and facial paralysis in children. Ann Otol Rhinol Laryngol 105: 58-62. doi: 10.1177/000348949610500110
    [26] Spiegel JH, Lustig LR, Lee KC, et al. (1998) Contemporary presentation and management of a spectrum of mastoid abscesses. Laryngoscope 108: 822-828. doi: 10.1097/00005537-199806000-00009
    [27] Marioni G, de Filippis C, Tregnaghi A, et al. (2001) Bezold's abscess in children: Case report and review of the literature. Int J Pediatr Otorhinolaryngol 61: 173-177. doi: 10.1016/S0165-5876(01)00564-X
    [28] Vlastos I, Helmis G, Athanasopoulos I, et al. (2010) Acute mastoiditis complicated with bezold abscess, sigmoid sinus thrombosis and occipital osteomyelitis in a child. Eur Rev Med Pharmacol Sci 14: 635-638.
    [29] Nelson D, Jeanmonod R (2013) Bezold abscess: A rare complication of mastoiditis. Am J Emerg Med 31: 1626. e1623-1626. e1624.
    [30] Al-Baharna H, Al-Mubaireek H, Arora V (2016) Bezold's abscess: A case report and review of cases over 14 years. Indian J Otology 22: 148. doi: 10.4103/0971-7749.187978
    [31] Govea-Camacho LH, Pérez-Ramírez R, Cornejo-Suárez A, et al. (2016) Diagnosis and treatment of the complications of otitis media in adults. Case series and literature review. Cirugía Y Cirujanos (English Edition) 84: 398-404.
    [32] McMullan B (2009) Bezold's abscess: A serious complication of otitis media. J Paediatr Child Health 45: 616-618. doi: 10.1111/j.1440-1754.2009.01575.x
    [33] Minor LB, Glasscock ME, Poe D (2010)  Glasscock-Shambaugh Surgery of the ear PMPH-USA.
    [34] Yildirim I, Shea KM, Pelton SI (2015) Pneumococcal disease in the era of pneumococcal conjugate vaccine. Infect Dis Clin 29: 679-697. doi: 10.1016/j.idc.2015.07.009
    [35] Yorgancılar E, Yıldırım M, Gun R, et al. (2013) Complications of chronic suppurative otitis media: A retrospective review. Eur Arch Otorhinolaryngol 270: 69-76. doi: 10.1007/s00405-012-1924-8
    [36] Secko M, Aherne A (2013) Diagnosis of Bezold abscess using bedside ultrasound. J Emerg Med 44: 670-672. doi: 10.1016/j.jemermed.2012.07.066
  • This article has been cited by:

    1. Adriana Cătaş, On the Fekete–Szegö Problem for Meromorphic Functions Associated with p,q-Wright Type Hypergeometric Function, 2021, 13, 2073-8994, 2143, 10.3390/sym13112143
  • Reader Comments
  • © 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(6308) PDF downloads(110) Cited by(1)

Figures and Tables

Figures(2)

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog