Recent zbMATH articles in MSC 30C45https://www.zbmath.org/atom/cc/30C452022-05-16T20:40:13.078697ZWerkzeugHankel, Toeplitz, and Hermitian-Toeplitz determinants for certain close-to-convex functionshttps://www.zbmath.org/1483.300322022-05-16T20:40:13.078697Z"Allu, Vasudevarao"https://www.zbmath.org/authors/?q=ai:allu.vasudevarao"Lecko, Adam"https://www.zbmath.org/authors/?q=ai:lecko.adam"Thomas, Derek K."https://www.zbmath.org/authors/?q=ai:thomas.derek-keithSummary: Let \(f\) be analytic in \(\mathbb{D}=\{z\in \mathbb{C} : |z|<1\}\), and be given by \(f(z) = z+\sum_{n=2}^\infty a_nz^n\). We give sharp bounds for the second Hankel determinant, some Toeplitz, and some Hermitian-Toeplitz determinants of functions in the class of Ozaki close-to-convex functions, together with a sharp bound for the Zalcman functional \(J_{2, 3}(f)\).Certain estimates of normalized analytic functionshttps://www.zbmath.org/1483.300332022-05-16T20:40:13.078697Z"Anand, Swati"https://www.zbmath.org/authors/?q=ai:anand.swati"Jain, Naveen Kumar"https://www.zbmath.org/authors/?q=ai:jain.naveen-kumar"Kumar, Sushil"https://www.zbmath.org/authors/?q=ai:kumar.sushilSummary: Let \(\phi\) be a normalized convex function defined on open unit disk \(\mathbb{D}\). For a unified class of normalized analytic functions which satisfy the second order differential subordination \(f'(z)+\alpha zf''(z)\prec\varphi(z)\) for all \(z\in\mathbb{D}\), we investigate the distortion theorem and growth theorem. Further, the bounds on initial logarithmic coefficients, inverse coefficient and the second Hankel determinant involving the inverse coefficients are examined.Paatero's \(V(k)\) space and a claim by Pinchukhttps://www.zbmath.org/1483.300342022-05-16T20:40:13.078697Z"Andreev, Valentin V."https://www.zbmath.org/authors/?q=ai:andreev.valentin-v"Bekker, Miron B."https://www.zbmath.org/authors/?q=ai:bekker.miron-b"Cima, Joseph A."https://www.zbmath.org/authors/?q=ai:cima.joseph-aSummary: In this article we obtain a factorization theorem for the functions in Paatero's \(V(k)\) space. We bring attention to a significant result of Pinchuk which unfortunately is false. This result relates measures associated to functions in \(V(k)\) and an integral representation theorem for such functions. We prove necessary and sufficient conditions for a wide class of functions (in particular, the polynomials) to belong to the Paatero class based on the geometry of their critical points, and obtain explicit representation of the measures associated to a wide class of such polynomials that includes the Suffridge polynomials.Coefficient estimates for Libera type bi-close-to-convex functionshttps://www.zbmath.org/1483.300352022-05-16T20:40:13.078697Z"Bulut, Serap"https://www.zbmath.org/authors/?q=ai:bulut.serapSummary: In a very recent paper, \textit{Z. Wang} and the author [C. R., Math., Acad. Sci. Paris 355, No. 8, 876--880 (2017; Zbl 1376.30015)]
determined the estimates for the general Taylor-Maclaurin coefficients of functions belonging to the bi-close-to-convex function class. In this study, we introduce the class of Libera type bi-close-to-convex functions and obtain the upper bounds for the coefficients of functions belonging to this class. Our results generalize the results in the above mentioned paper.Convolution conditions for two subclasses of analytic functions defined by Jackson \(q\)-difference operatorhttps://www.zbmath.org/1483.300362022-05-16T20:40:13.078697Z"El-Emam, Fatma Z."https://www.zbmath.org/authors/?q=ai:el-emam.fatma-zSummary: By using Jackson \(q\)-derivative, some characterizations in terms of convolutions for two classes of analytic functions in the open unit disc are given. Also, coefficient conditions and inclusion properties for functions in these classes are found.A proof of Hall's conjecture on length of ray images under starlike mappings of order \(\alpha\)https://www.zbmath.org/1483.300372022-05-16T20:40:13.078697Z"Hästö, Peter"https://www.zbmath.org/authors/?q=ai:hasto.peter-a"Ponnusamy, Saminathan"https://www.zbmath.org/authors/?q=ai:ponnusamy.saminathanSummary: Assume that \(f\) lies in the class of starlike functions of order \(\alpha\in[0,1)\), that is, which are regular and univalent for \(|z|< 1\) and such that
\[
\mathrm{Re}\left(\frac{zf'(z)}{f(z)}\right)> \alpha\quad\text{for } |z|<1.
\]
In this paper we show that for each \(\alpha\in[0,1)\), the following sharp inequality holds:
\[
|f(re^{i\theta})|^{-1}\int_0^r|f'(ue^{i\theta})|\,du\leq\frac{\Gamma(\frac{1}{2})\Gamma(2-\alpha)}{\Gamma(\frac{3}{2}-\alpha)} \quad\text{for every } r< 1 \text{ and } \theta.
\]
This settles the conjecture of \textit{R. R. Hall} [Bull. Lond. Math. Soc. 12, 119--126 (1980; Zbl 0442.30007)] positively.Littlewood-Paley conjecture associated with certain classes of analytic functionshttps://www.zbmath.org/1483.300382022-05-16T20:40:13.078697Z"Kumar, Virendra"https://www.zbmath.org/authors/?q=ai:kumar.virendra"Srivastava, Rekha"https://www.zbmath.org/authors/?q=ai:srivastava.rekha"Cho, Nak Eun"https://www.zbmath.org/authors/?q=ai:cho.nakeun|cho.nak-eunSummary: The Littlewood-Paley conjecture hardly holds for any subclass of univalent functions except the class of starlike functions as verified, in general, by the researchers until now. Therefore, it is interesting to consider the classes where the Littlewood-Paley conjecture holds completely or partially. For such investigation, the classes of normalized strongly \(\alpha\)-close-to-convex functions and \(\alpha\)-quasiconvex functions of order \(\beta\) are considered in this paper. In the main, bounds on the initial coefficients and related Fekete-Szegö inequalities are derived in this paper. Furthermore, it is seen that the Littlewood-Paley conjecture holds for all values of the parameter \(\gamma >0\) in case of the first coefficient. However for the second coefficient, it holds for large positive values of \(\gamma\). Relevant connections of our results with the existing results are also pointed out.Certain subclasses of univalent functions involving Pascal distribution serieshttps://www.zbmath.org/1483.300392022-05-16T20:40:13.078697Z"Lashin, Abdel Moneim Y."https://www.zbmath.org/authors/?q=ai:lashin.abdel-moneim-yousof"Badghaish, Abeer O."https://www.zbmath.org/authors/?q=ai:badghaish.abeer-o"Bajamal, Amani Z."https://www.zbmath.org/authors/?q=ai:bajamal.amani-zSummary: In this paper, our aim is to find the necessary and sufficient conditions for univalent functions involving Pascal distribution to be in some subclasses of analytic functions.On a subclass of analytic functions that are starlike with respect to a boundary point involving exponential functionhttps://www.zbmath.org/1483.300402022-05-16T20:40:13.078697Z"Lecko, Adam"https://www.zbmath.org/authors/?q=ai:lecko.adam"Murugusundaramoorthy, Gangadharan"https://www.zbmath.org/authors/?q=ai:murugusundaramoorthy.gangadharan"Sivasubramanian, Srikandan"https://www.zbmath.org/authors/?q=ai:sivasubramanian.srikandanSummary: In the present exploration, the authors define and inspect a new class of functions that are regular in the unit disc \(\mathfrak{D}:=\{\varsigma \in \mathbb{C} : |\varsigma| < 1\}\), by using an adapted version of the interesting analytic formula offered by Robertson (unexploited) for starlike functions with respect to a boundary point by subordinating to an exponential function. Examples of some new subclasses are presented. Initial coefficient estimates are specified, and the familiar Fekete-Szegö inequality is obtained. Differential subordinations concerning these newly demarcated subclasses are also established.Fourth Hankel determinant for a subclass of starlike functions based on modified sigmoidhttps://www.zbmath.org/1483.300412022-05-16T20:40:13.078697Z"Mashwani, Wali Khan"https://www.zbmath.org/authors/?q=ai:mashwani.wali-khan"Ahmad, Bakhtiar"https://www.zbmath.org/authors/?q=ai:ahmad.bakhtiar"Khan, Nazar"https://www.zbmath.org/authors/?q=ai:khan.nazar"Khan, Muhammad Ghaffar"https://www.zbmath.org/authors/?q=ai:khan.muhammad-ghaffar"Arjika, Sama"https://www.zbmath.org/authors/?q=ai:arjika.sama"Khan, Bilal"https://www.zbmath.org/authors/?q=ai:khan.bilal"Chinram, Ronnason"https://www.zbmath.org/authors/?q=ai:chinram.ronnasonSummary: In our present investigation, we obtain the improved third-order Hankel determinant for a class of starlike functions connected with modified sigmoid functions. Further, we investigate the fourth-order Hankel determinant, Zalcman conjecture, and also evaluate the fourth-order Hankel determinants for 2-fold, 3-fold, and 4-fold symmetric starlike functions.A family of holomorphic functions defined by differential inequalityhttps://www.zbmath.org/1483.300422022-05-16T20:40:13.078697Z"Mohammed, Nafya Hameed"https://www.zbmath.org/authors/?q=ai:hameed-mohammed.nafya"Adegani, Ebrahim Analouei"https://www.zbmath.org/authors/?q=ai:adegani.ebrahim-analouei"Bulboacă, Teodor"https://www.zbmath.org/authors/?q=ai:bulboaca.teodor"Cho, Nak Eun"https://www.zbmath.org/authors/?q=ai:cho.nak-eunSummary: The aim of the present paper is to introduce and study a subfamily of holomorphic and normalized functions defined by a differential inequality. Some geometric properties of this family of holomorphic functions and different problems of a family of such functions are presented.Fekete-Szegö inequality for certain subclasses of analytic functions related with crescent-shaped domain and application of poison distribution serieshttps://www.zbmath.org/1483.300432022-05-16T20:40:13.078697Z"Murugusundaramoorthy, Gangadharan"https://www.zbmath.org/authors/?q=ai:murugusundaramoorthy.gangadharanSummary: The purpose of this paper is to define a new class of analytic, normalized functions in the open unit disk \(\mathbb{D}=\{ z:z\in \mathbb{C}\text{ and } \left\vert z\right\vert <1\}\) subordinating with crescent shaped regions, and to derive certain coefficient estimates \(a_2, a_3\) and Fekete-Szegö inequality for \(f\in\mathcal{M}_q(\alpha,\beta,\lambda)\). A similar result have been done for the function \(f^{-1}\). Further application of our results to certain functions defined by convolution products with a normalized analytic function is given, in particular we obtain Fekete-Szegö inequalities for certain subclasses of functions defined through Poisson distribution series.On an extension of Nunokawa's lemmahttps://www.zbmath.org/1483.300442022-05-16T20:40:13.078697Z"Nunokawa, Mamoru"https://www.zbmath.org/authors/?q=ai:nunokawa.mamoru"Sokół, Janusz"https://www.zbmath.org/authors/?q=ai:sokol.januszSummary: Jack's Lemma says that if \(f(z)\) is regular in the disc \(|z|\le r\), \(f(0)=0\), and \(|f(z)|\) assumes its maximum at \(z_0\) on the circle \(|z|=r\), then \(z_0f'(z)_0/f(z_0)\ge 1\). This Lemma was generalized in several directions. In this paper we consider an improvement of some first author's results of this type.Coefficients of the inverse of functions for the subclass of the class \(\mathcal{U} (\lambda)\)https://www.zbmath.org/1483.300452022-05-16T20:40:13.078697Z"Obradović, Milutin"https://www.zbmath.org/authors/?q=ai:obradovic.milutin"Tuneski, Nikola"https://www.zbmath.org/authors/?q=ai:tuneski.nikolaSummary: Let \(\mathcal{A}\) be the class of functions \(f\) that are analytic in the unit disk \(\mathbb{D}\) and normalized such that \(f(z)=z+a_2z^2+a_3z^3+\cdots\). Let \(0<\lambda \leq 1\) and
\[
\begin{aligned} \mathcal{U} (\lambda) = \left\{f\in \mathcal{A} : \left| \left(\frac{z}{f(z)} \right)^2f'(z)-1\right| < \lambda, z\in \mathbb{D}\right\}. \end{aligned}
\]
In this paper sharp upper bounds of the first three coefficients of the inverse function \(f^{-1}\) are given in the case when
\[
\begin{aligned} \frac{f(z)}{z}\prec \frac{1}{(1-z)(1-\lambda z)}. \end{aligned}
\]Properties of functions with symmetric points involving subordinationhttps://www.zbmath.org/1483.300462022-05-16T20:40:13.078697Z"Raza, Malik Ali"https://www.zbmath.org/authors/?q=ai:raza.malik-ali"Bukhari, Syed Zakar Hussain"https://www.zbmath.org/authors/?q=ai:bukhari.syed-zakar-hussain"Ahmed, Imtiaz"https://www.zbmath.org/authors/?q=ai:ahmed.imtiaz"Ashfaq, Muhammad"https://www.zbmath.org/authors/?q=ai:ashfaq.muhammad"Nazir, Maryam"https://www.zbmath.org/authors/?q=ai:nazir.maryamSummary: We study a new subclass of functions with symmetric points and derive an equivalent formulation of these functions in term of subordination. Moreover, we find coefficient estimates and discuss characterizations for functions belonging to this new class. We also obtain distortion and growth results. We relate our results with the existing literature of the subject.On sufficient conditions for strongly starlikeness of order a and type \(\beta\)https://www.zbmath.org/1483.300472022-05-16T20:40:13.078697Z"Sharma, Vidyadhar"https://www.zbmath.org/authors/?q=ai:sharma.vidyadhar"Mathur, Nisha"https://www.zbmath.org/authors/?q=ai:mathur.nisha"Soni, Amit"https://www.zbmath.org/authors/?q=ai:soni.amitSummary: By making use of differential subordination technique, we derive certain conditions for \(p\)-valent strongly starlike functions of order \(\alpha\) and type \(\beta\). The results presented here are sharp.Properties of analytic solutions of three similar differential equations of the second orderhttps://www.zbmath.org/1483.300482022-05-16T20:40:13.078697Z"Sheremeta, M. M."https://www.zbmath.org/authors/?q=ai:sheremeta.myroslav-m"Trukhan, Yu. S."https://www.zbmath.org/authors/?q=ai:trukhan.yu-sSummary: An analytic univalent in \(\mathbb{D}=\{z:\;|z|<1\}\) function \(f(z)\) is said to be convex if \(f(\mathbb{D})\) is a convex domain. It is well known that the condition \(\operatorname{Re}\{1+zf''(z)/f'(z)\}>0\), \(z\in\mathbb{D} \), is necessary and sufficient for the convexity of \(f\). The function \(f\) is said to be close-to-convex in \(\mathbb{D}\) if there exists a convex in \(\mathbb{D}\) function \(\Phi\) such that \(\operatorname{Re}(f'(z)/\Phi'(z))>0\), \(z\in\mathbb{D} \).
S.M. Shah indicated conditions on real parameters \(\beta_0\), \(\beta_1\), \(\gamma_0\), \(\gamma_1\), \(\gamma_2\) of the differential equation \(z^2w''+(\beta_0 z^2+\beta_1 z)w'+(\gamma_0z^2+\gamma_1 z+\gamma_2) w=0\), under which there exists an entire transcendental solution \(f\) such that \(f\) and all its derivatives are close-to-convex in \(\mathbb{D} \).
Let \(0<R\le+\infty\), \(\mathbb{D}_R=\{z:\;|z|<R\}\) and \(l\) be a positive continuous function on \([0,R)\), which satisfies \((R-r)l(r)>C\), \(C=\operatorname{const}>1\). An analytic in \(\mathbb{D}_R\) function \(f\) is said to be of bounded \(l\)-index if there exists \(N\in\mathbb{Z}_+\) such that for all \(n\in\mathbb{Z}_+\) and \(z\in\mathbb{D}_R\)
\[\frac{|f^{(n)}(z)|}{n!l^n(|z|)}\le \max\left\{\frac{|f^{(k)}(z)|}{k!l^k(|z|)}:\;0\le k\le N\right\}.\]
Here we investigate close-to-convexity and the boundedness of the \(l\)-index for analytic in \(\mathbb{D}\) solutions of three analogues of Shah differential equation: \(z(z-1) w''+\beta z w'+\gamma w=0\), \((z-1)^2 w''+\beta z w'+\gamma w=0\) and \((1-z)^3 w''+\beta(1- z) w'+\gamma w=0\). Despite the similarity of these equations, their solutions have different properties.A note on spirallike functionshttps://www.zbmath.org/1483.300492022-05-16T20:40:13.078697Z"Sim, Y. J."https://www.zbmath.org/authors/?q=ai:sim.young-jong"Thomas, D. K."https://www.zbmath.org/authors/?q=ai:thomas.derek-keithSummary: Let \(f\) be analytic in the unit disk \(\mathbb{D}=\{z\in \mathbb{C}:|z|<1 \}\) and let \(\mathcal{S}\) be the subclass of normalised univalent functions with \(f(0)=0\) and \(f'(0)=1\), given by \(f(z)=z+\sum_{n=2}^{\infty }a_n z^n\). Let \(F\) be the inverse function of \(f\), given by \(F(\omega )=\omega +\sum_{n=2}^{\infty }A_n \omega^n\) for \(|\omega |\le r_0(f)\). Denote by \(\mathcal{S}_p^{*}(\alpha )\) the subset of \(\mathcal{S}\) consisting of the spirallike functions of order \(\alpha\) in \(\mathbb{D} \), that is, functions satisfying
\[ \operatorname{Re} \left\{e^{-i\gamma}\dfrac{zf'(z)}{f(z)}\right\}>\alpha\cos \gamma, \]
for \(z\in \mathbb{D}\), \(0\le \alpha <1\) and \(\gamma \in (-\pi /2,\pi /2)\). We give sharp upper and lower bounds for both \(|a_3|-|a_2|\) and \(|A_3|-|A_2|\) when \(f\in \mathcal{S}_p^{* }(\alpha )\), thus solving an open problem and presenting some new inequalities for coefficient differences.A certain subclass of analytic functions with negative coefficients defined by Gegenbauer polynomialshttps://www.zbmath.org/1483.300502022-05-16T20:40:13.078697Z"Venkateswarlu, Bolineni"https://www.zbmath.org/authors/?q=ai:venkateswarlu.bollineni"Reddy, Pinninti Thirupathi"https://www.zbmath.org/authors/?q=ai:reddy.pinninti-thirupathi"Sridevi, Settipalli"https://www.zbmath.org/authors/?q=ai:sridevi.settipalli"Sujatha, Vaishnavy"https://www.zbmath.org/authors/?q=ai:sujatha.vaishnavySummary: In this paper, we introduce a new subclass of analytic functions with negative coefficients defined by Gegenbauer polynomials. We obtain coefficient bounds, growth and distortion properties, extreme points and radii of starlikeness, convexity and close-to-convexity for functions belonging to the class \(TS_\lambda^m(\gamma,\varrho, k, \vartheta)\). Furthermore, we obtained the Fekete-Szego problem for this class.A conjecture on Marx-Strohhäcker type inclusion relation between \(q\)-convex and \(q\)-starlike functionshttps://www.zbmath.org/1483.300512022-05-16T20:40:13.078697Z"Verma, Sarika"https://www.zbmath.org/authors/?q=ai:verma.sarika"Kumar, Raj"https://www.zbmath.org/authors/?q=ai:kumar.raj"Sokół, Janusz"https://www.zbmath.org/authors/?q=ai:sokol.januszSummary: We prove that the class \(\mathcal{K}\) of normalized univalent convex functions defined in the unit disk \(\mathbb{E}\) is contained in \(\mathcal{K}_q\left(\frac{1-q}{1+q^2}\right)\) (\(0<q<1\)), the class of normalized univalent \(q\)-convex functions of order \((1-q)/(1+q^2)\). We provide examples that exhibit a Marx-Strohhäcker type inclusion relation, i.e. \(\mathcal{K}_q\left(\frac{1-q}{1+q^2}\right)\subset\mathcal{S}_q^\ast\left(\frac{1}{1+q}\right)\), where \(\mathcal{S}_q^\ast\left(\frac{1}{1+q}\right)\) is the class of \(q\)-starlike functions of order \(1/(1+q)\). Note that for \(q\to 1^-\) this relation coincides with the well-known result, \(\mathcal{K}\subset\mathcal{S}^\ast\left(\frac{1}{2}\right)\), of \textit{A. Marx} [Math. Ann. 107, 40--67 (1932; JFM 58.0363.01)]
and \textit{E. Strohhäcker} [M. Z. 37, 356--380 (1933; JFM 59.0353.02)].On the third and fourth Hankel determinants for a subclass of analytic functionshttps://www.zbmath.org/1483.300522022-05-16T20:40:13.078697Z"Wang, Zhi-Gang"https://www.zbmath.org/authors/?q=ai:wang.zhigang"Raza, Mohsan"https://www.zbmath.org/authors/?q=ai:raza.mohsan"Arif, Muhammad"https://www.zbmath.org/authors/?q=ai:arif.muhammad"Ahmad, Khurshid"https://www.zbmath.org/authors/?q=ai:ahmad.khurshidSummary: The objective of this paper is to investigate the third and fourth Hankel determinants for the class of functions with bounded turning associated with Bernoulli's lemniscate. The fourth Hankel determinants for 2-fold symmetric and 3-fold symmetric functions are also studied.Open-door lemma for functions with fixed second coefficientshttps://www.zbmath.org/1483.300572022-05-16T20:40:13.078697Z"Amani, M."https://www.zbmath.org/authors/?q=ai:amani.mostafa"Aghalary, R."https://www.zbmath.org/authors/?q=ai:aghalary.rasoul"Ebadian, A."https://www.zbmath.org/authors/?q=ai:ebadian.aliSummary: We extend the well-known open-door lemma by using the theory of differential subordination for functions with fixed initial coefficient and apply it to study some integral operators. Further, using an extension of Nunokawa lemma, we determine some sufficient conditions for the radii and order of starlike functions with fixed second coefficient. Our results improve and generalize some previously known results.On the coefficients of certain subclasses of harmonic univalent mappings with nonzero polehttps://www.zbmath.org/1483.310012022-05-16T20:40:13.078697Z"Bhowmik, Bappaditya"https://www.zbmath.org/authors/?q=ai:bhowmik.bappaditya"Majee, Santana"https://www.zbmath.org/authors/?q=ai:majee.santanaSummary: Let \(Co(p), p\in (0,1]\) be the class of all meromorphic univalent functions \(\varphi\) defined in the open unit disc \(\mathbb{D}\) with normalizations \(\varphi (0)=0=\varphi^{\prime} (0)-1\) and having simple pole at \(z=p\in (0,1]\) such that the complement of \(\varphi (\mathbb{D})\) is a convex domain. The class \(Co(p)\) is called the class of concave univalent functions. Let \(S_H^0 (p)\) be the class of all sense preserving univalent harmonic mappings \(f\) defined on \(\mathbb{D}\) having simple pole at \(z=p\in (0,1)\) with the normalizations \(f(0)=f_z (0)-1=0\) and \(f_{\bar{z}}(0)=0\). We first derive the exact regions of variability for the second Taylor coefficients of \(h\) where \(f=h+\overline{g}\in S_H^0 (p)\) with \(h-g\in Co(p)\). Next we consider the class \(S_H^0 (1)\) of all sense preserving univalent harmonic mappings \(f\) in \(\mathbb{D}\) having simple pole at \(z=1\) with the same normalizations as above. We derive exact regions of variability for the coefficients of \(h\) where \(f=h+\overline{g}\in S_H^0 (1)\) satisfying \(h-e^{2i\theta}g\in Co(1)\) with dilatation \(g^{\prime} (z)/h^{\prime} (z)=e^{-2i\theta}z\), for some \(\theta, 0\leq \theta <\pi\).Some properties of certain close-to-convex harmonic mappingshttps://www.zbmath.org/1483.310072022-05-16T20:40:13.078697Z"Wang, Xiao-Yuan"https://www.zbmath.org/authors/?q=ai:wang.xiaoyuan"Wang, Zhi-Gang"https://www.zbmath.org/authors/?q=ai:wang.zhigang|wang.zhigang.1"Fan, Jin-Hua"https://www.zbmath.org/authors/?q=ai:fan.jinhua"Hu, Zhen-Yong"https://www.zbmath.org/authors/?q=ai:hu.zhenyongSummary: In this paper, we determine the sharp estimates for Toeplitz determinants of a subclass of close-to-convex harmonic mappings. Moreover, we obtain an improved version of Bohr's inequalities for a subclass of close-to-convex harmonic mappings, whose analytic parts are Ma-Minda convex functions.Spacetimes with continuous linear isotropies. I: Spatial rotationshttps://www.zbmath.org/1483.830062022-05-16T20:40:13.078697Z"MacCallum, M. A. H."https://www.zbmath.org/authors/?q=ai:maccallum.malcolm-a-hSummary: The weakest known criterion for local rotational symmetry (LRS) in spacetimes of Petrov type D is due to \textit{S. W. Goode} and \textit{J. Wainwright} [ibid. 18, 315--331 (1986; Zbl 0584.53029)]. Here it is shown, using methods related to the Cartan-Karlhede procedure, to be equivalent to local spatial rotation invariance of the Riemann tensor and its first derivatives. Conformally flat spacetimes are similarly studied and it is shown that for almost all cases the same criterion ensures LRS. Only for conformally flat accelerated perfect fluids are three curvature derivatives required to ensure LRS, showing that Ellis's original condition for that case is necessary as well as sufficient.Spacetimes with continuous linear isotropies. III: Null rotationshttps://www.zbmath.org/1483.830202022-05-16T20:40:13.078697Z"MacCallum, M. A. H."https://www.zbmath.org/authors/?q=ai:maccallum.malcolm-a-hSummary: It is shown that in many of the possible cases local null rotation invariance of the curvature and its first derivatives is sufficient to ensure that there is an isometry group \(G_r\) with \(r\ge 3\) acting on (a neighbourhood of) the spacetime and containing a null rotation isotropy. The exceptions where invariance of the second derivatives is additionally required to ensure this conclusion are Petrov type N Einstein spacetimes, spacetimes containing ``pure radiation'' (a Ricci tensor of Segre type [(11,2)]), and conformally flat spacetimes with a Ricci tensor of Segre type [1(11,1)] (a ``tachyon fluid'').
For Parts I and II, see [the author, ibid. 53, No. 6, Paper No. 57, 21 p. (2021; Zbl 1483.83006); ibid. 53, No. 6, Paper No. 61, 12 p. (2021; Zbl 1483.83019)].