Cowling, V. F.

Royster, Wimberly C.

English

University of Kentucky

University of KentuckyUKnowledgeMathematics Faculty Publications Mathematics1-1961Some Applications of the Weierstrass Mean ValueTheoremV. F. CowlingUniversity of KentuckyWimberly C. RoysterUniversity of KentuckyRight click to open a feedback form in a new tab to let us know how this document benefits you.Follow this and additional works at: https://uknowledge.uky.edu/math_facpubPart of the Mathematics CommonsThis Article is brought to you for free and open access by the Mathematics at UKnowledge. It has been accepted for inclusion in Mathematics FacultyPublications by an authorized administrator of UKnowledge. For more information, please contact UKnowledge@lsv.uky.edu.Repository CitationCowling, V. F. and Royster, Wimberly C., "Some Applications of the Weierstrass Mean Value Theorem" (1961). Mathematics FacultyPublications. 2.https://uknowledge.uky.edu/math_facpub/2Some Applications of the Weierstrass Mean Value TheoremNotes/Citation InformationPublished in Journal of the Mathematical Society of Japan, v. 13, no. 1, p. 104-108.The copyright holder has granted the permission for posting the article here.Digital Object Identifier (DOI)http://dx.doi.org/10.2969/jmsj/01310104This article is available at UKnowledge: https://uknowledge.uky.edu/math_facpub/2J. Math. Soc. JapanVol. 13, No. 1, 1961Some applications of the Weierstrassmean value theoremBy V. F. COWLING and W. C. ROYSTER(Received Sept. 12, 1959)1. Introduction. Let $f(z)$ be an analytic function holomorphic in some do-main $D$ of the complex plane. Then $f(z)$ is said to be univalent in $D$ if for anytwo points in $D$ of affix $z_{1}$ and $z_{2}$ it is true that $f(z_{1})=f(z_{2})$ implies $z_{1}=z_{2}$ . Nowin order that $f(z)$ be univalent in $D$ it is necessary, but not sufficient, that $f^{\prime}(z)$$\neq 0$ for $z\in D$ . If, however, we specialize $D$ and require that $D$ be convex then$Re\{f^{\prime}(z)\}>0,$ $z\in D$ , implies that $f(z)$ is univalent in $D$ . This latter result wasfirst proved by J. Wolff [7] for the case in which $D$ is a half-plane and gener-alized by Noshiro and Warschawski [2], [6] to convex domains. On the otherhand Herzog and Piranian [1] demonstrated that if a domain $D$ has the pro-perty that $f(z)$ is univalent in $D$ whenever ${\rm Re}\{f^{\prime}(z)\}>0$ throughout $D$ thenthe domain is not far from being convex. Thus if we propose to obtain condi-tions for univalency in non-convex domains, we must expect some restrictionson $f^{\prime}(z)$ other than ${\rm Re}\{f^{\prime}(z)\}>0$ . In this note some conditions of this natureare given, bearing mostly on $Argf^{\prime}(z)$ for $z\in D$ . These conditions arise in anatural manner from an application of a theorem due to Weierstrass [5]. Theresults are similar to those obtained by Reade [3] whose results are based upona criterion for univalency due to Umezawa [4]. The theorem of Weierstrassemployed is the followingTHEOREM (Weierstrass). Let $C$ be a rectifiable curve joining the poinls of affix$ z=\alpha$ and $ z=\beta$ . Let $z=z(t)(a\leqq t\leqq b)$ be a parametric representation of C. $TheIl$if $f(z)$ is continuous on $C$ and $g(z)$ is contimtous and positive on $C$$\int_{a^{b}}f(z)g(z)dt=X\int_{a^{b}}g(z)dt$where the point of affix $X$ is contained in the convex hull of the set of $va[lJ$eswhich $f(z)$ assumes on $C$.2. Applications of the Weierstrass theorem. For the sake of completenesswe begin with a proof of the Noshiro-Warschawski Theorem.THEOREM (Noshiro-Warschawski). A function regular in a convex domain $D$is univalent in $D$ if ${\rm Re}\{f^{\prime}(z)\}>0$ for $z\in D$ .PROOF. Let $z_{1}$ and $z_{2}$ be two distinct points of $D$ . Then upon integratingthe derivative of $f(z)$ along the line segment connecting $z_{1}$ and $z_{2}$ we haveApplications of the Weierstrass mean value theorem 105$f(z_{2})-f(z_{1})=\int_{z_{2}^{z_{1}}}f^{\prime}(z)dz$$=(z_{2}-z_{1})\int_{0^{1}}f^{\prime}(z_{2}+(z_{1}-z_{1})t)dt$ .But by the Weierstrass theorem the latter integral has the value $Z^{*}$ , where$Z^{*}$ is the affix of a point lying in the convex hull $C^{*}$ of the image of the linesegment $\overline{z_{1}z_{2}}$ under the mapping effected by $f^{\prime}(z)$ . But since $Re\{f^{\prime}(z)\}>0$ for$z\in D$ it follows that the origin is not in $C^{*}$ and hence $f(z_{2})-f(z_{1})=Z^{*}(z_{2}-z_{1})$$\neq 0$ unless $z_{2}=z_{1}$ .REMARK. Clearly the theorem is still valid if for some $0\leqq\alpha\leqq 2\pi,$ ${\rm Re}\{e^{-i\alpha}f^{\prime}(z)\}$$>0$ for $z\in D$ .We next prove a theorem first given by Reade [3]. Our proof differs fromReade’s in that whereas Reade employed a condition for univalency due toUmezawa we use the mean value theorem of Weierstrass.THEOREM 2.1. Let $\varphi$ be fixed and $ 0\leqq\varphi\leqq\pi$ . Let $D$ be a domain in which itis possible to join each pair of distinct points $z_{1},$ $z_{3}$ by a pair of straight line seg-ments $\overline{z_{1}z_{2}},\overline{z_{2}z_{3}}$ lying in $D$ such that(1)$|Arg\frac{z_{3}-z_{2}}{z_{2}-z_{1}}|\leqq\varphi$$\varphi$ being independent of $z_{1}$ and $z_{3}$ in D. Then if $f(z)$ is analytic in $D$ and if forsome $ 0\leqq\alpha\leqq 2\pi$$\alpha\leqq Argf^{\gamma}(z)\leqq\pi-\varphi+\alpha$for $z\in D,$ $f(z)$ is univalent in $D$ .PROOF. Let $z_{1}$ and $z_{3}$ be two distinct points of $D$ and let $z_{2}\neq z_{1},$ $\neq z_{3}$ bea point of $D$ such that the line segments $\overline{z_{1}z_{2}}$ and $\overline{z_{2}z_{3}}$ lie in $D$ and satisfy (1).Now we may write$f(z_{3})-f(z_{1})=\int_{z}^{z_{l^{1}}}f^{\prime}(z)dz+\int_{z}^{z_{2^{S}}}f^{\prime}(z)dz$$=(z_{2}-z_{1})\int_{0^{1}}f^{\prime}(z_{1}+(z_{2}-z_{1})t)dt$$+(z_{3}-z_{2})\int_{0^{1}}f^{\prime}(z_{2}+(z_{3}-z_{2})t)dt$ .Hence by the Weierstrass mean value theorem we have(2) $f(z_{3})-f(z_{1})=Z_{1^{*}}(z_{2}-z_{1})+Z_{1}^{*}(z_{8}-z_{2})$$=Z_{1^{*}}(z_{2}-z_{1})[1+\frac{Z_{2^{*}}}{Z_{1^{*}}}z_{2}^{3}-z_{1}^{2}z-z]$ ,since by hypothesis, $|Arg(Z_{2^{*}}/Z_{1^{*}})|<\pi-\varphi$ it follows from (1) that106 V. F. COWLING and W. C. ROYSTER$|Arg\frac{Z_{2^{*}}}{Z_{1^{*}}}\frac{z_{3}-z_{2}}{z_{2}-z_{1}}|<\pi$ .Therefore the quantity in the brackets in the last member of (2) cannot vanishand, by (2), $f(z_{3})\neq f(z_{1})$ .THEOREM 2.2. Let $D$ be a domain any two points of which may be joined bythe arc (lying in $D$) of an ellipse $z=z_{0}+e^{i\beta}(a\cos t+ib\sin t)$ , for which $t_{1}\leqq t\leqq t_{2}$ ,$0\leqq t_{1},$ $t_{2}\leqq\frac{\pi}{2}$ , and where $a,$ $b>0,$ $\beta$ real. ($a,$ $b,$ $\beta$ may depend on the two pointsto be connected.) Then if $f(z)$ is analytic in $D$ and if for some $ 0\leqq\alpha\leqq 2\pi$$\alpha<Argf^{\prime}(z)<\alpha+\frac{\pi}{2}$for all $z\in D,$ $f(z)$ is univalent in $D$ .PROOF. Let $z_{1}$ and $z_{2}$ be two points of $D$ then$f(z_{2})-f(z_{1})=\int_{z^{z_{1^{2}}}}f^{\prime}(z)dz$$=e^{t\beta}\int_{t^{t_{\iota^{2}}}}f^{\prime}(z)(-a\sin t+ib\cos t)dt$ ,where $z_{k}=z_{0}+e^{i\beta}$ (a $\cos t_{k}+ib\sin t_{k}$), $k=1,2$ .Applying the Weierstrass theorem and simplifying yields(3) $f(z_{2})-f(z_{1})=e^{i\beta}[-a\int_{c^{t_{1}}}lf^{\prime}(z)\sin tdT+ib\int_{t^{t_{1}}}\dot{f}^{\prime}(z)\cos tdt]$$=Z_{2^{*}}e^{i\beta}[\frac{Z_{1}^{*}}{Z_{2^{*}}}{\rm Re}\{e^{-i\beta}(z_{2}-z_{1})\}+i{\rm Im}\{e^{-i\beta}(z_{2}-z_{1})\}]$ ,where $Z_{1}^{*}$ and $Z_{2^{*}}$ are points in the convex hull of the sets which are imagesof the elliptic arcs from $z_{1}$ to $z_{2}$ under the mapping effected by $f^{\prime}(z)$ . It re-sults from the hypothesis that $|Arg(Z_{1^{*}}/Z_{2^{*}})|<\frac{\pi}{2}$ and hence, since $Z^{*}\neq 0$ ,the right hand member of (3) is never zero unless $z_{2}-z_{1}$ so that $f(z)$ is uni-valent in $D$ .DEFINITION. A domain $D$ is said to have property $U$ with constant $\theta=\theta(D)$if when $z_{1}$ and $z_{2}$ in $D$ are given there exists a constant $\theta$ (independent of $z_{1}$and $z_{2}$), $\theta<\pi$, and a sequence of points $z_{1}=\zeta_{0},$ $\zeta_{1},$ $\zeta_{2}$ , $\cdot$ , $\zeta_{n}=z_{2}$ with the pro-perty that the segments $\overline{\zeta_{0}\zeta_{1}},$ $\overline{\zeta_{1}\zeta_{2}},$ $\cdots$ , $\overline{\zeta_{n-1}\zeta_{n}}$ all belong to $D$ and$|Arg\frac{\zeta_{k+1}-\zeta_{k}}{z_{2}-z_{1}}|\leqq\theta$ , $k=0,1,$ $\cdots$ , $n-1$ .REMARK. A convex domain has the property $U$ with constant $\theta=0$ .THEOREM 2.3. Let $D$ be a domain having the property U. Then if for some$0\leqq\alpha\leqq 2\pi,$ $Argf^{\prime}(z)$ satisfies $\alpha+\theta<Argf^{\prime}(z)<\pi+\alpha-\theta,$ $\theta<\frac{\pi}{2}$ , for $z$ in $D;f(z)$Applications of the Weierstrass mean value theorem 107is univalent in $D$ .PRCOF. We may write$f(z_{\gamma,\lrcorner})-f(z_{1})=\sum_{k=0}^{n-1}\int_{\zeta^{\zeta_{k^{k+1}}}}f^{\prime}(z)dz$$=(z_{2}-z_{1})\sum_{k=0}^{n-1}\lambda_{k}\frac{\zeta_{k+1}-\zeta_{k}}{z_{2}-z_{1}}$where the vectors $\lambda_{k}$ satisfy $\alpha+\theta<Arg\lambda_{k}<\pi+\alpha-\theta$ for some $ 0\leqq\alpha\leqq 2\pi$ . Thedomain possesses the property $U$ and we have$\alpha<Arg\lambda_{k}\frac{\zeta_{k+1}-\zeta_{k}}{z_{2}-z_{1}}<\alpha+\pi$ , $k=0,1,$ $\cdots,$ $n-1$ ,and consequently$\sum_{k=0}^{71-1}\lambda_{k}\frac{\zeta_{k+1}-\zeta_{k}}{z_{2}-z_{1}}\neq 0$which implies $f(z)$ is univalent.The following interesting theorem is obtained if $\theta(D)=\frac{\pi}{2}$ .THEOREM 2.4. Let $D$ be a domain any two points of which may be joined bya finite number of line segments (lying in $D$) and parallel to the coordinate axissuch that the $x$ and $y$ coordinates of the end points form either a non-decreasingor non-increasing sequence. $T/\iota en$ if for some $ 0\leqq\alpha\leqq 2\pi$$\alpha<Argf^{\prime}(z)<\alpha+\frac{\pi}{2}$for $z\in D,$ $f(z)$ is univalent in $D$ .The proof follows along the same lines of Theorem 2.3 and 2.1.We can easily construct examples for which Theorem 2.23. Examples.and 2.3 apply but for which Reade’s theorem does not apply. For example con-sider the region in the first quadrant bounded by two confocal ellipses withcenter at the origin and with foci at $\pm 1$ . If the difference between theircorresponding semi axes is sufficiently small the region does not satisfy thecriterion of Reade’s theorem. Theorem 2.2 or 2.3 applied to the function$f(z)=z\sin^{-1}z+\sqrt{1-z^{3}}$ in this region shows that $f(z)$ is univalent in the region.University KentuckyBibliography[1] F. Herzog and G. Piranian, On the univalence of functions whose derivative hasa positive real part, Proc. Amer. Math. Soc., 2 (1951), 625-633.[2] K. Noshiro, On the theory of schlicht functions, J. Fac. Sci. Hokkaido Imp. Univ.,108 V. F. COWLING and W. C. ROYSTERSapparo (I), 2 (1934-1935), 129-155.[3] M. Reade, On Umezawa’s criteria for univalence II, J. Math. Soc. Japan, 10 (1958),255-259.[4] T. Umezawa, On the theory of univalent functions, T\^ohoku Math. J. 7 (1955),212-228.[5] G. Valiron, Th\’eorie des fonctions, Paris, 1948, p. 362.[6] S. Warschawski, On the higher derivatives at the boundary in conformal mapping,Trans. Amer. Math. Soc., 38 (1935), 310-340.[7] J. Wolff, L’int\’egral d’une fonction holomorphe et \‘a partie r\’eele positive dansune demi-plan est univalent, C. R. Acad. Sci. Paris, 198 (1934), 1209-1210.

Some Applications of the Weierstrass Mean Value Theorem

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Some Applications of the Weierstrass Mean Value Theorem

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