Isac G., Nemeth S. Scalar and asymptotic scalar derivatives.. Theory and applications (Springer, 2008)(ISBN 0387739874)(O)(252s) MCat

This book is devoted to the study of scalar and asymptotic scalar derivativesand their applications to the study of some problems considered in nonlinearanalysis, in geometry, and in app

SCALAR AND ASYMPTOTIC SCALAR DERIVATIVES

Theory and Applications

J Birge (University of Chicago)

C.A Floudas (Princeton University)

F Giannessi (University of Pisa)

H.D Sherali (Virginia Polytechnic and State University)

T Terlaky (McMaster University)

Y Ye (Stanford University)

Aims and Scope

Optimization has been expanding in all directions at an astonishing rate during the last few decades New algorithmic and theoretical techniques have been developed, the diffusion into other disciplines has proceeded at a rapid pace, and our knowledge of all aspects of the field has grown even more profound At the same time, one of the most striking trends in optimization is the constantly increasing emphasis on the interdisciplinary nature of the field Optimization has been a basic tool in all areas of applied mathematics, engineering, medicine, economics and other sciences

The series Optimization and Its Applications publishes undergraduate

and graduate textbooks, monographs and state-of-the-art expository works that focus on algorithms for solving optimization problems and also study applications involving such problems Some of the topics covered include nonlinear optimization (convex and nonconvex), network flow problems, stochastic optimization, optimal control, discrete optimization, multi-objective programming, description of software packages, approximation techniques and heuristic approaches

SCALAR AND ASYMPTOTIC SCALAR DERIVATIVES

Theory and Applications

GEORGE ISAC

Royal Military College of Canada, Kingston, Ontario, Canada

SÁNDOR ZOLTÁN NÉMETH

University of Birmingham, Birmingham, United Kingdom

By

123

George Isac Sándor Zoltán Németh

Royal Military College of Canada

Library of Congress Control Number: 2007934547

¤ 2008 Springer Science+Business Media, LLC

All rights reserved This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY

10013, USA), except for brief excerpts in connection with reviews or scholarly analysis Use in connection with any form of information storage and retrieval, electronic adaptation, computer software,

or by similar or dissimilar methodology now known or hereafter developed is forbidden

The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject

Birmingham B15 2TT

He offers all to others, and his life

is more abundant He helps all men alike,

and his life is more exuberant

(Lao Zi: Truth and Nature)

This book is devoted to the study of scalar and asymptotic scalar derivativesand their applications to the study of some problems considered in nonlinearanalysis, in geometry, and in applied mathematics.

The notion of a scalar derivative is due to S Z N´emeth, and the notion of

an asymptotic scalar derivative is due to G Isac Both notions are recent, neverconsidered in a book, and have interesting applications About applications,

we cite applications to the study of complementarity problems, to the study

of fixed points of nonlinear mappings, to spectral nonlinear analysis, and tothe study of some interesting problems considered in differential geometry andother applications

A new characterization of monotonicity of nonlinear mappings is anotherremarkable application of scalar derivatives

A relation between scalar derivatives and asymptotic scalar derivatives, alized by an inversion operator is also presented in this book This relationhas important consequences in the theory of scalar derivatives, and in someapplications For example, this relation permitted us a new development of themethod of exceptional family of elements, introduced and used by G Isac incomplementarity theory

re-Now, we present a brief description of the contents of this book

Chapter 1 is dedicated to the study of scalar derivatives in Euclidean spaces

In this chapter we explain the reason for introducing scalar derivatives as goodmathematical tools for characterizing important properties of functions from

RntoRn In order to avoid some difficulties, we consider only upper and lowerscalar derivatives which are extensions to vector functions of Dini derivatives

We consider also the case when lower and upper scalar derivatives coincide

This is a strong restriction and we show that for n = 2 the existence of a

single-valued scalar derivative is strongly related to complex differentiability Thelower and upper scalar derivatives are also used to characterize convexity likenotions

Chapter 2 essentially has two parts In the first part we present the notion ofthe asymptotic derivative and some results related to this notion and in the secondpart we introduce the notion of the asymptotic scalar derivative The resultspresented in the first part are necessary for understanding the notions given inthe second part It is known that the notion of the asymptotic derivative wasintroduced by the Russian school, in particular by M A Krasnoselskii, underthe name of asymptotic linearity The main goal of this chapter is to present thenotion of the asymptotic scalar derivative and some of its applications.Chapter 3 presents the scalar derivatives in Hilbert spaces and several resultsand properties are given We note that in this chapter we give the definitions

of scalar derivatives of rank p, named briefly for p = 2, scalar derivatives We also put in evidence the fact that the case p = 1 is strongly related to the notion

of submonotone mapping, introduced in 1981 by J E Spingarn and studied in

1997 by P Georgiev Several new results related to computation of the scalarderivative and some interesting relations with skew-adjoint operators are alsopresented The scalar derivatives are used to characterize the monotonicity ofmappings in Hilbert spaces Many of the formulae presented in this chapter arise

from applications such as fixed point theorems, surjectivity theorems, integral

equations, and complementarity problems, among others

Chapter 4 contains the extension of the theory of scalar derivatives to Banachspaces This extension is based on the notion of the semi-inner product inLumer’s sense The notion of scalar derivatives defined in this case is applied tofixed point theory, to the study of solvability of integral equations, of variationalinequalities, and of complementarity problems

Chapter 5 is dedicated to a generalization of the notion of Kachurovskii–Minty–Browder monotonicity to Riemannian manifolds and to realize this weintroduce the notion of the geodesic monotone vector field The geodesic con-vexity for mappings is also considered For a global example of monotonevector fields we consider Hadamard manifolds (complete, simply connectedRiemannian manifolds with nonpositive sectional curvature) Analyzing theexistence of geodesic monotone vector fields, we prove that there are no strictlygeodesic monotone vector fields on a Riemannian manifold that contain a closedgeodesic We note that many results presented in this chapter are based on ageneralization to Riemannian manifolds of scalar derivatives studied in the pre-vious chapters The nongradient type monotonicity on Riemannian manifolds

is considered for the first time in a book

This book is the first book dedicated to the study of scalar and asymptoticscalar derivatives and certainly new developments related to these notions arepossible

It is impossible to finish this preface without giving many thanks to thepeople who spent their time developing the open source tools (operating sys-tem, window manager, and software) that were essential for writing this book,

greately reducing the time and energy spent in word processing These opensource tools are: the Linux and FreeBSD operating systems, the Ratpoison win-dow manager, the LaTeX word processing language, and the VIM and Bluefisheditors.

We are grateful to the reviewers for their valuable comments and tions Taking them into consideration has greately improved the quality andpresentation of the book

sugges-To conclude, we would like to say that we very much appreciated the excellentassistance offered to us by the staff of Springer Publishers

1 Scalar Derivatives in Euclidean Spaces 11.1 Scalar Derivatives of Mappings in Euclidean Spaces 11.1.1 Some Basic Results Concerning Skew-Adjoint

1.1.2 The Scalar Derivative and its Fundamental Properties 31.1.3 Case n = 2 The Relation of the Scalar Derivative

1.1.4 Miscellanea Concerning Scalar Differentiability 91.1.5 Characterization of Monotonicity by Scalar Derivatives 12

1.2.1 Scalar Derivatives and Directional Derivatives 15

1.3 Monotonicity, Scalar Differentiability, and Conformity 241.3.1 The Coefficient of Conformity and the Conformal

2 Asymptotic Derivatives and Asymptotic Scalar Derivatives 31

2.3 Asymptotic Differentiability Along a Convex Cone in a

2.4 Asymptotic Differentiability in Locally Convex Spaces 49

3 Scalar Derivatives in Hilbert Spaces 79

3.5 Variational Inequalities and Complementarity Problems 97

3.6.4 Infinitesimal Exceptional Family of Elements 106

3.7.2 Exceptional Family of Elements for an Ordered Pair

3.7.3 Infinitesimal Exceptional Family of Elements for an

3.8.6 Infinitesimal Exceptional Family of Elements 125

3.9 The Asymptotic Browder–Hartman–Stampacchia

Condition and Interior Bands of ε-Solutions for Nonlinear

3.9.1 Preliminaries 134

3.9.3 The asymptotic Browder–Hartman–Stampacchia

3.10 REFE-Acceptable Mappings and a Necessary and Sufficient

Condition for the Nonexistence of Regular Exceptional

3.10.2 Mappings Without Regular Exceptional Family of

Elements A necessary and Sufficient Condition 157

4.5.1 A Fixed Point Index for α-condensing Mappings 166

4.5.4 Applications of Krasnoselskii-Type Fixed Point

4.5.5 Applications of Altman-Type Fixed Point Theorems 175

5 Monotone Vector Fields on Riemannian Manifolds

5.1.1 Geodesic Monotone Vector Fields and Convex

5.1.2 Geodesic Monotone Vector Fields and the First

5.1.3 Closed Geodesics and Geodesic Monotone

5.1.4 The Geodesic Monotonicity of Position Vector Fields 185

5.2 Killing Monotone Vector Fields 2005.2.1 Expansive One-Parameter Transformation Groups 2005.2.2 Geodesic Scalar Derivatives and Conformity 206

5.3.1 Some Basic Consequences of the Comparison

5.3.3 Projection maps generating monotone vector fields 214

Scalar Derivatives in Euclidean Spaces

1.1 Scalar Derivatives of Mappings in Euclidean Spaces

The behaviour of the scalar productf(x) − f(y), x − y (with f : R n → R n

and ,  the usual scalar product in R n ) when x and y run overRnis a good tool

in characterizing important properties of f If f is bounded, then this product converges to 0 for x → y Therefore it cannot be used in obtaining a local

characterization Hence it is natural to consider at y limits of the expressions

of the form f(x) − f(y), x − y/x − y, x − y for x → y Thus we

arrive naturally at a notion that we call the scalar derivative It is in general amultivalued mapping fromRntoR even if f is linear.

In order to avoid the difficulties in considering multifunctions we only sider so-called upper and lower scalar derivatives, which are extensions to vectorfunctions of the Dini derivatives

con-We consider mostly the case when lower and upper scalar derivatives cide This restriction is a very strong one In Section 1.1.3 it is shown that

coin-for n = 2 the existence of a single-valued scalar derivative is strongly related

to the complex differentiability In Section 1.1.4 we consider various ples and counterexamples Lower and upper scalar derivatives can be used incharacterizing the monotone operators in the way this is done in Section 1.1.5.Convex functionals have as gradients monotone operators Hence the scalarderivative can also be used to characterize convexity like notions Thus Propo-sitions 2.1 and 2.2 in Karamardian and Schaible [1990] together with the results

exam-in our Section 1.1.5 give some characterizations of convex and strictly convexfunctionals

We have defined the notion of scalar derivative having in mind Minty’s tonicity notion [Minty, 1962] To simplify the notations, in this chapter a mono-

mono-tone mapping (strictly monomono-tone mapping) f will be called increasing (strictly

increasing) If−f is monotone (strictly monotone), then f will be called

de-creasing (strictly dede-creasing)

1.1.1 Some Basic Results Concerning Skew-Adjoint

Operators

Definition 1.1 Consider the operator f : Rn → R n It is called increasing

(decreasing) if for any x and y inRn one has

f(x) − f(y), x − y ≥ 0 (≤ 0).

If

f(x) − f(y), x − y > 0 (< 0)

whenever x = y, then f is called strictly increasing (strictly decreasing).

Definition 1.2 The linear operator A : Rn → R n is called skew-adjoint if for any x and y inRn the relation Ax, y + Ay, x = 0 holds.

Theorem 1.3 If A : Rn → R n is linear, then the following statements are equivalent.

1 A is skew-adjoint.

2 Ax − Ay, x − y = 0 for any x, y ∈ R n

3 Taking an arbitrary orthonormal basis inRn , A can be represented by a

matrix A = (a ij)i,j=1, ,n such that a ij =−a ji ∀i, j ∈ {1, 2, , n}.

Proof 1 ⇒ 2 Take x and y arbitrarily in R n By the definition of the

skew-adjoint operator A we have Ax, y + Ay, x = 0 Put y = x Then

Ax, x = 0 for arbitrary x in R n Whence we also haveAx−Ay, x−y = 0

by the linearity of A The implication 2 ⇒ 1 can be shown similarly.

The equivalence 1⇔ 3 is obvious 

Remark 1.1

1 There exist injective skew-adjoint operators For instance, the operators

represented by the matrices

2 If n is odd, then there is no injective skew-adjoint operator inRn Indeed let A be the matrix corresponding to an skew-adjoint operator Let the

superscript T denote transposition Then A T = −A and hence det A =

−det A this means that det A = 0.

Theorem 1.4 Consider the operator F : Rn → R n The following tions are equivalent.

asser-1 F (x) − F (y), x − y = 0, ∀x, y ∈ R n

2 F is an affine operator with a skew-adjoint linear term.

Proof Suppose that 1 holds Put f (x) = F (x) − F (0) for x in R n Then

f (0) = 0 and f(x) − f(y), x − y = 0 ∀x, y ∈ R n Let x be arbitrary inRn

and y = 0 Then f(x), x = 0 ∀x ∈ R n The above relation also yields

f(x), x − f(x), y − f(y), x + f(y), y = 0, ∀x, y ∈ R n

and hence

f(x), y + f(y), x = 0, ∀x, y ∈ R n

Put x = λx1+ μx2with arbitrary x1and x2inRn Then

f(λx1+ μx2), y  = −f(y), λx1+ μx2 = −λf(y), x1 − μf(y), x2

= λ f(x1), y  + μf(x2), y ,

wherefrom

f(λx1+ μx2)− λf(x1)− μf(x2), y  = 0

for any x1, x2and y inRn and any λ, μ inR, wherefrom we have the linearity of

f Because f(x)−f(y), x−y = 0, for any x, y in R n , f is also skew-adjoint Thus F (x) = f (x) + F (0) and hence it is indeed affine with a skew-adjoint

linear term

The implication 2⇒ 1 is obvious 

1.1.2 The Scalar Derivative and its Fundamental

exists (here x−x0 2=x−x0, x −x0), then it is called the scalar derivative

of the operator f in x0 In this case f is said to be scalarly differentiable at x0.

If f#(x) exists for every x inRn , then f is said to be scalarly differentiable on

Rn , with the scalar derivative f#.

It follows from this definition that both the set of operators scalarly differentiable

in x0, and the set of operators scalarly differentiable onRnform linear spaces.Definition 1.6 Consider the operator f : Rn → R n The limit

Theorem 1.7 The linear operator A : Rn → R n is scalarly differentiable

onRn if and only if it is of the form A = B + cI n with B skew-adjoint linear

operator, I n the identity ofRn , and c a real number.

Proof Let us suppose that A is scalarly differentiable in x0 ∈ R n Then

Theorem 1.8 Suppose that f : Rn → R n , f = (f1, · · · , f n ) is scalarly

differentiable in x0 Then for every i ∈ {1, , n} there exists the partial

Theorem 1.9 Suppose that f : Rn → R n is differentiable in x0and scalarly differentiable in x0 Then we have for the differential df (x0) of f at x0 the relation

df (x0) = B + f#(x0)I n ,

with B : Rn → R n linear and skew-adjoint

Proof Let t ∈ R nbe given Then

1 The theorem holds for the Gateaux differential δf (x0) in place of df (x0).

The differentiability condition is often used next and hence we state the theorem for this stronger condition.

2 If we denote by f  (x0) the Jacobi matrix of f at x0in some coordinate

repre-sentation and the symbols B and I n stand for matrices of the corresponding operators, then our relation becomes

f  (x

0) = B + f#(x0)I n

Theorem 1.10 Suppose that f : Rn → R n , f = (f1, , f n ) is

differen-tiable in x0 Then the following statements are equivalent.

Condition 2 is called the Cauchy–Riemann relation at x0

Proof 1⇒ 2 By Remark 2 one has

f  (x

0) = B + f#(x0)I n ,

where B is a skew-symmetric matrix and f  (x0) is the Jacobi matrix of f at x0.

Because from the above relation

.

.

.

1.1.3 Casen = 2 The Relation of the Scalar Derivative

with the Complex Derivative

We identify in this chapter the complex numbers with points inR2 The scalarproduct of these numbers means the scalar product of the vectors representingthem inR2

Theorem 1.11 Let f : C → C be a complex function The following

statements are equivalent.

1 f is differentiable in z0as a complex function.

2 f is differentiable in z0as a mapping f : R2 → R2 and is scalarly entiable in this point.

The differentiability condition of f at z0 in 2 is essential In examples 2and 3 of Section 1.1.4 we construct two discontinuous mappings at 0, whichare scalarly differentiable in this point

Remark 1.3

1 Let G be an open subset of C Then f is holomorphic on G if and only if it

is differentiable as a vector function and scalarly differentiable on G As is

well known, the set of holomorphic functions on C is closed with respect to

the compositions of functions.

2 The above remark justifies the following generalization of a holomorphic

function.

Definition 1.12 Let G be open inRn The mapping f : Rn → R n is called

R-holomorphic on G if and only if it is differentiable and scalarly differentiable

on G The set of R-holomorphic mappings on G is denoted by H(G).

Theorem 1.13 For the complex function f : C → C the following

state-ments are equivalent

1 f is differentiable in z0∈ C as a complex function.

2 f and if are scalarly differentiable in z0.

Proof Let us denote f = u + iv, z = x + iy, z0 = x0+ iy0 Then

Remark 1.4 The function f is holomorphic on the open set G ⊂ C if and

only if f and if are scalarly differentiable on G.

Theorem 1.14 If f : C → C is differentiable in z0 as a complex function then f and if are scalarly differentiable in z0and the relation

1.1.4 Miscellanea Concerning Scalar Differentiability

Examples and counterexamples

1 Let n ∈ N; n > 2 Then the set H(R n) of the holomorphic functions on

Rnis not closed under compositions of functions

Indeed consider A : Rn → R n represented by the matrix A = (a ij)i,j =

Obviously, A is a skew-adjoint operator Hence A is holomorphic onRn

Consider A2= A ◦ A and assume that it is holomorphic Then by Theorem

1.7 it must be of the form A2 = B + cI n with B a skew-adjoint linear operator, c a real number, and I n the identical map Let us denote the

matrix representing A2by D = (d ij)i,j=1, ,n , then d12+ d21= 0 That is

(a11a12+· · · + a 1n a n2 ) + (a21a11+· · · + a 2n a n1 ) = 0.

From the definition of A it follows that 2(n − 2) = 0 and hence n = 2,

contradicting the hypothesis on n.

From this example and the results in Section 1.1.3 the next assertionfollows

Theorem 1.15 The set of scalarly differentiable linear mappings inRn

is closed under composition if and only if n ≤ 2.

From the definition of the scalar derivative the next assertion followseasily

Lemma 1.16 Let 0 = (0, , 0) ∈ R n and let f : Rn → R n be a mapping having the properties:

(a) f (0) = 0.

(b) f(x), x = 0, ∀x ∈ R n

Then f is scalarly differentiable in 0 and f#(0) = 0

Usage of this lemma allows us to construct the following two examples ofdiscontinuous mappings at 0, which are scalarly differentiable in this point

where A is a nonzero, linear, skew-adjoint operator.

In fact, Example 3 generalizes Example 2 Both mappings satisfy the ditions of the above lemma and hence they are scalarly differentiable in 0

con-Let us show that f in 3 is not continuous at 0 Because A = 0, there exists

some t inRn with At = 0 Put x = λt, λ > 0 Then the relation

shows that f is not continuous at 0.

4 Example of a mapping f : R2 → R2 which is continuous at 0, scalarlydifferentiable in this point, but not differentiable as a vector function

The continuity of the two components of f is a standard exercise in calculus.

If f is differentiable in 0, then its components f1and f2are differentiablereal-valued functions Because

∂f1(0, 0)

∂f1(0, 0)

then df1(0, 0) = 0 Hence we cannot have

lim inf

x→0 y→0

that is, the Cauchy–Riemann conditions do not hold at 0

The scalar differentiability of f at 0 follows from the fact that it satisfies

the conditions of the lemma

6 Example of a mapping f : R2 → R2which is continuous at 0, satisfies theCauchy–Riemann conditions at 0, but is not scalarly differentiable at thispoint

Assume that f is scalarly differentiable in 0 Then the limit

must exist Put x = 0 and y → 0 Then this limit will be 0 Put x = y = 0,

x → 0 Then this limit will be 1/2 That is, the limit does not exist.

7 Example of a nonlinearR-holomorphic mapping f : R n → R nfor arbitrary

pings of other type) Because H(R n ) for n > 2 is not closed with respect to

the composition (see Example 1 and Theorem 1.15), we cannot derive other holomorphic mappings in this way.

1.1.5 Characterization of Monotonicity by Scalar

Proof The implication 1⇒ 2 is obvious.

2⇒ 1 Take ε > 0 arbitrarily and put g = f + εI n Then

g#(x) = f#(x) + ε > 0, ∀x ∈ G.

Take a, b in G; a = b For x in the line segment [a, b] determined by a and b,

one has by hypothesis:

lim inf

y →x

g(y) − g(x), y − x

y − x 2 > 0,

and hence there exists δ(x) > 0 such that for any y in I x =]x −δ(x)(b−a), x+ δ(x)(b − a)[⊂ G, g(y) − g(x), y − x > 0 holds as far as y = x Obviously,

for an appropriate set y1, , y m −1 of points in ]a, b[ We can suppose that

y1, , y m −1 are ordered from a to b Hence a = y0 ∈ I y1, b = y m ∈ I y m −1

We can also consider that no interval I y i is contained in any other Take ξ i ∈

I y i −1 ∩ I y i ∩]y i −1 , y i[ Then by the construction of these intervals

The case f#(x) ≤ 0, ∀x ∈ G can be handled similarly. Theorem 1.18 Let G be an open convex set in Rn and suppose that

f : G → R n satisfies

f#(x) > 0 (f#(x) < 0), ∀x ∈ G.

Then f is strictly increasing (strictly decreasing) on G.

The proof of this theorem is in fact contained in the proof of Theorem 1.17.Corollary 1.19 Let f : Rn → R n be given The following statements are equivalent.

1 f#(x) = 0, ∀x ∈ R n

2 f is an affine mapping with a skew-adjoint linear part.

Proof The implication 2⇒ 1 is trivial.

To show that 1⇒ 2 we apply Theorem 1.17 to conclude that

Proof 1⇒ 2 Suppose that f(y) − f(x), y − x ≥ 0 ∀x, y ∈ R n Take

y = x + λt with λ ∈ R, λ > 0, and t ∈ R narbitrarily Then

2⇒ 1 Suppose that the Gateaux differential is positive semi-definite at each

point of G Take a, b in G, a = b and x ∈ [a, b] Then δf(x)t, t ≥ 0 for

every t ∈ R n; that is,

Take t = b − a and y = x + λt, then

where ab denotes the line determined by a and b By an appropriate adaptation

of the method in the proof of Theorem 1.17 we can conclude

f(b) − f(a), b − a ≥ 0.

Theorem 1.21 Let G be a convex open subset ofRn If f : G → R n is Gateaux differentiable on G and the Gateaux differential in each x of G is posi-

tive definite (negative definite), then f is strictly increasing (strictly decreasing)

on G.

Proof Suppose that the Gateaux differential is positive definite Let a, b in

G, a = b Let x ∈ [a, b] Then δf(x)t, t > 0 for every t ∈ R n \{0}; that is,

Take t = b − a and y = x + λt Then

lim infy→x

y ∈ab

f(y) − f(x), y − x

y − x 2 > 0,

where ab denotes the line determined by a and b Reasoning similar to that in

the proof of Theorem 1.17 yields that

f(b) − f(a), b − a > 0.



1.2 Computational Formulae for the Scalar Derivative

In this section we consider relations of the scalar derivatives with the directionalderivatives and with the basic notions of spectral theory In the two-dimensionalcase we exhibit a geometric connection with the Kasner circle The obtainedformulae are used for determining the monotonicity domains of operators on

Rn

1.2.1 Scalar Derivatives and Directional Derivatives

Let f : Rn → R n be given and x, h ∈ R n If the limit

f  (x; h) = lim inf

t ↓0

1

t (f (x + th) − f(x))

exists with t in R, then it is called the directional derivative of f at x in the direction h To have a geometrical sense we suppose that h = 0, but f  (x; 0) =

0 can be taken obviously for each x.

The operator f is called locally Lipschitz at x, if there exist a neighbourhood

V of x and a positive real number L such that for any y and z in V the inequality

f(y) − f(z) ≤ L y − z ... the set of operators scalarly differentiable

in x0, and the set of operators scalarly differentiable onRnform linear spaces.Definition 1.6 Consider... if and< /i>

only if f and if are scalarly differentiable on G.

Theorem 1.14 If f : C → C is differentiable in z0 as a complex function then f and. .. (f#(x) < 0) if and only if df (x) is positive definite (negative

definite).

3 f is scalarly differentiable in x and f#(x) = if and only if df (x)