 Alright, so we proved this Zalkman's lemma in the last lecture and you know basically it is a theorem and it is a characterization of normal families and I also remarked that you know at the end of the lecture I remarked that the converse of Zalkman's lemma is also true therefore Zalkman's lemma is a characterization of you know non-normal families okay and it is like and you know it is like Marty's theorem, Marty's theorem is a characterization of normal families of Meromorphic functions and Zalkman's lemma is a characterization of non-normal families okay and both of them are very important in the when you go to the proof of the Picard theorem which is our main aim okay. So I just want to begin with the following remark, so recall that you know when I was giving you the motivation for Zalkman's lemma I actually proved this result, so if you take a family script f of Meromorphic functions defined on a domain in the complex plane or even in the external plane and you take a point Z0 in the domain where the family is normal okay then you give then this normality at a point which is supposed to mean normality in some neighbourhood of the point, so normality at a point is defined just like analyticity at a point is defined okay, so normality at a point means normality in some open disc surrounding that point okay and if you have a family is normal at a point then it has this property that you know given any sequence of points tending to that point is it tending to Z0 and a sequence of decreasing positive radii, a sequence of radii tending to 0 okay then give me any sequence in the family I can find a subsequence such that the zoomed functions converge normally to a constant function on the plane okay this is what we proved and this was very easy to deduce okay and Zalkman's lemma actually tells you that if the family is not normal you will get the exact opposite namely you will be able to find a sequence such that the zoomed functions converge to a non-constant Meromorphic function okay that is the big difference okay and what I want to tell you is that the converse of this proposition is also true if you apply if you it is a simple it is an exercise which you might for example do you will have to use a diagonalization argument okay and you can do this simple exercise and well not so simple but not so hard also but you have to use Zalkman's lemma and show that the converse of this proposition is true okay so I will write that down first so here is a theorem so here is again a proposition this is a converse of the proposition stated before Zalkman's lemma and what is this proposition well it is a criterion for normality. So suppose script F is a family of Meromorphic functions on a neighbourhood of a point Z naught okay the point Z naught could be a point in the extended plane really could even be the point at infinity does not matter okay suppose that for every sequence Z n tending to Z naught every sequence epsilon n tending to 0 plus epsilon n are all positive numbers okay given any sequence f n in the family script F there exists a subsequence f n k such that the zoom sequence so what is the zoom sequence it is g is g n k of zeta is the zooming of f n k at z k with the zooming factor 1 by epsilon k and using the variable zeta and this is just f n k is of z k plus epsilon k times zeta okay so this zoom sequence converges normally on the complex plane to a constant value in so this constant value can also be the point at the value infinity so it is a constant value in the extended plane okay. So suppose this property is satisfied by the family script F okay that whenever you give me a sequence of points converging to Z naught and a sequence of radii going to 0 from any sequence f n I can extract a subsequence for the zoom sequence converges normally to a constant okay then script F has to be normal at zeta okay then script F has to be normal at zeta that means it has to be normal in some open neighborhood of zeta okay and I want to write down the details but I leave it as an exercise to you proof is use Salkman's lemma and a diagonalization argument assuming the assuming the family script F is not normal not normal on a decreasing sequence of neighborhoods of z naught. So I will just this is actually more often this is an exercise okay this is an exercise that I want you to do so the proof is by contradiction okay. So you assume I have to show that the family is normal at z naught which means I have to show that the family is normal in some open neighborhood of z naught. So if that is not true it means that in every open neighborhood of z naught the family is not normal. So you take a decreasing sequence of open neighborhoods of z naught okay the neighborhoods becoming smaller and smaller for example you can take decreasing a decreasing sequence of open disk centered at z naught with radii 1 by n where n goes to infinity okay and on each of these disks you can apply Salkman's lemma because Salkman's lemma applies to a non-normal family okay and then you will get sequences in from the Salkman's lemma and then you apply it the right diagonal you apply a diagonalization argument and then apply the hypothesis of the proposition and you will get a contradiction okay. So I leave it to you to do that right. So well so now let me continue with the main result. The main result is well you know what we are going to which is going to be the main one should say this is really the deep theorem that is the theorem of Montel okay and it is a theorem on normality okay and it is a deep theorem because it involves lots of things it involves several theorems in complex analysis and it is the key to proving the Picard theorems okay. So the point is that so this is the Montel theorem and what is the Montel theorem? The Montel theorem it is actually a it is a normality it is a theorem for normality of a family and what it says is that you take a family of meromorphic functions on a domain okay if you know that the functions in the family always omit 3 values in the extended plane 3 distinct values in the extended plane then the family is normal okay. So it is a beautiful theorem to check that a family is normal all you make sure is that you find 3 values in the extended plane which means you have to find 2 complex finite values in the complex plane the finite complex plane and 1 probably the value infinity okay. So you should somehow find 3 values that all the functions in the family miss okay and they should omit these values and if you do that then you can then the theorem says that the family is normal okay. So somehow omission of values is connected to normality okay the normality of the family is connected to omission of certain number of values of by functions in the family okay and all theorem says that if you can make sure that the family omits 3 values then you are sure it is normal okay and mind your normality is a condition for compactness okay. So you can imagine the theorem is very powerful you are saying that a certain family of meromorphic functions on a domain you are saying that it is compact in the sense that you know it is normally sequentially compact that is every sequence admits a normally convergent sequence okay to that is a very strong property okay and to deduce that all you need to say you have to just verify that the family omits 3 values okay 3 values and it is really beautiful. So in particular what it means is that you know if you take a family of analytic functions on a domain okay if you take a family of analytic functions on a domain okay and if it and if you know that it omits 2 finite complex values okay then you can immediately say that it is normal okay because when you are looking at analytic functions you can forget the third value which is infinity okay infinity is already omitted okay. So this version of the theorem mortal theorem is called the fundamental normality test okay so if you want to check a family of analytic functions is normal okay on a domain all you have to do is you make sure that it omits make sure that every member of the family omits 2 fixed complex values okay and then you are sure that it is normal okay. So it is a very deep theorem so let me state it so this is Montel's theorem on normality function so here is the theorem let script F be a family of Meromorphic functions on a domain D in the extended complex plane such that script F omits 3 distinct values in C union infinity. So what does this mean that is there are elements lambda 1, lambda 2, lambda 3 in C union infinity and of course they are all distinct lambda i not equal to lambda j for i not equal to j okay such that for each F in the family F does not take the values any of the values lambda 1, lambda 2, lambda 3 these are omitted values then script F is normal so this is the I mean the deepest I would say this is the deepest theorem in this course okay this is the most important theorem in this course alright. To check that a family of Meromorphic functions on a domain is normal you just ensure that it omits 3 values all the functions in the family omit 3 fixed values okay 3 distinct fixed values alright so here is so how do we go about the proof so the first thing I want to tell you is that you know fundamental property of Mobius transformations that you know given any 3 values in the extended plane you can find a Mobius transformation that can map those 3 values to 0, 1 and infinity okay so you can always I mean this is the way in which you write down a Mobius transformation in terms of cross ratios because you know a Mobius transformation has a fundamental property that it preserves cross ratios so this is something that you should have seen in the first course in complex analysis. So you know these 3 values lambda 1, lambda 2 and lambda 3 in the extended plane you know I can apply a Mobius transformation and make those map those values to 0, 1 and infinity okay and then I can compose the whole family by this I can transform the family using this Mobius transformation this I transform the whole family by using this Mobius transformation and therefore without loss of generality by using a Mobius transformation I can assume that the values that are omitted the 3 distinct values that are omitted are 0, 1 and infinity okay so this is the first reduction right so let me write this down using a Mobius transformation we may assume without loss of generality lambda 1 is 0 lambda 2 is 1 lambda 3 is equal to infinity okay you can do this right so you can assume and of course by Mobius transformation I mean a bilinear transformation or linear fractional transformation okay so that is the first thing then the second thing is that you know the domain on which the domain D in the extended plane where this family is defined okay that domain also can be you can change that domain and scale it so that you know it is it contains the unit disk okay okay so the point is that see I am trying what am I trying to check I am trying to check that this family is normal my final aim is to check the family is normal but how do I check it is normal I check it is normal by checking it is normal at every point because normal normality at a point means normality in a open small open disks containing that point and of and the property of being normal is a local property so if you check it at every point that is if you check it in an open abode of every point that is enough to check it is normal on the whole domain okay so it is like checking analyticity you do not have to check if you want to check a functions analytic on a whole domain it is enough to check at every point of the domain it is analytic so what I will have to do is I will have to I can assume that I am checking normality of the family on a small disk on a domain which is like a disk okay and of course you know if I am checking at the point at infinity alright then I will have to take a neighbourhood of infinity which is exterior of a disk in the complex plane and I will have to change the variable from z to 1 by z to make it into a disk surrounding the origin so in any case I can I can always assume that I am checking normality on a disk in the complex plane and I can translate the disk to the origin and scale it so that it contains the disk contains unit disk okay so again this is another reduction I am making without loss of generality I will it is enough for me to check that the family is defined on the unit disk okay so this is another reduction okay without loss of generality we may assume that D contains the unit disk mod z less than 1 okay because basically because you have to check normality locally and that means you have to check normality on a disk surrounding every point and that disk I can assume it to be the unit disk okay because I can always translate any small disk to the origin okay so that the centre of the small disk goes to the origin and then I can scale it so that it is big enough so that it contains the unit disk okay and translations and scalings are also mobius transformations so they are not going to modify the properties of the family okay. So I can so my situation is like this I now have a family of meromorphic functions okay I now have a family of meromorphic functions defined on the unit disk and what is given to me is that they omit the value 0, 1 and infinity I have to check that the family is normal okay but look at the beauty of it since these functions omit the value infinity they are analytic okay because you know a meromorphic function takes the value infinity only at a pole and the moment you assume that it does not take the value infinity all the functions have become analytic alright and the other thing is that all the functions are non-vanishing also because the value 0 is omitted okay so you are having non-vanishing analytic functions on the unit disk okay and they all omit the and what is the nice thing now is that you know if you have a non-vanishing analytic function on a simply connected domain you can always find k throats okay which are analytic okay because the reason is if you have a non-vanishing analytic function on a simply connected domain you can find a logarithm for the function and once you find a logarithm multiplying that analytic logarithm by 1 by k and then taking exponential okay so e power 1 by k log will give you a k th root of the function which is analytic so the advantage now is that your family has k th roots every function in your family has k th roots for all k okay and the trick is what the point is that since the see if a function does not take the value 1 okay then its k th root cannot take the value which is equal to a k th root of unity if the original function does not take a value 1 then its k th root cannot the analytic k th root cannot take a k th root of unity as a value okay and what case we will be using we will be using 2 power case okay so I am going to write that down note that script f is analytic and non-vanishing on mod set less than 1 which is simply connected so the so you know so the families so I will put this f sub k this is f to the 1 by 2 to the k where f is in script f are defined and analytic on the unit disk with omission of the values 0, infinity and 1 by 1 by 2 to the k which are 2 power k th roots of unity okay so I am so you see this mind you the whole point is f to the 1 by 2 power k is defined as e to the 1 by 2 to the k log f and this log f an analytic branch of log f exists because the domain is the unit disk it is simply connected and f never vanishes on the domain okay so this is something that is very important okay so fine and you know see now you but where are now you know I want you to understand what the idea is see the idea is you know these so you look at these functions okay these functions are defined on the unit disk okay but then you know what I am trying to prove that they are all normal I am trying to prove that this family script f is normal on the unit disk okay and mind you that means that I can extract the normality is just that I can extract from any sequence normally convergent subsequence okay but you see if I can extract such a from a sequence a normally convergent subsequence I can do that also for the 2 power k th roots okay so it is obvious that you know the family script f is normal if and only if any of the families script f sub k is normal for any k greater than 1 okay so actually it amounts to to show to showing that f is normal it is enough to show that one of the script f case is normal okay and therefore if you contradict the normality of script f what happens is you are contradicting in one stroke the normality is of each of the script f sub case okay and once you contradict the normality of each of the script f sub case Zalkman's Lemma comes into the picture and gives you a zoomed limit function which is a non-constant meromorphic function on C with spherical derivative equal to 1 at the origin and the spherical derivative is always bounded by 1 okay and the beautiful thing is that function that you get is an entire function okay because it is a limit of functions from each of these families it will not take the value infinity so it will be analytic and it will be defined on the whole complex plane so it will be entire okay and then you will see that you can get a contradiction easily by applying levels there okay so let me write this down note that f is normal normal if and only if fk is normal for every k or for some k okay so we are going to proceed by contradiction what we will do is assume f script f is not normal okay because I want we would like to use Zalkman's Lemma which is a characteristic of non-normality alright so assume f is not normal thus f sub k is not normal for every k alright and Zalkman's Lemma lemma gives for each script fk a zoomed limit function I will call the zoomed limit function as g sub k of zeta okay on the whole complex plane okay which is meromorphic non-constant its spherical derivative at the origin is 1 and all its spherical derivatives are bounded by 1 okay this is what Zalkman's Lemma tells you see Zalkman's Lemma tells you that whenever a family is not normal I can get a zoomed limit function which is non-constant meromorphic and the non-constancy is kind of normalized or fixed by making the spherical derivative to be 1 at the origin and mind you the limit function is defined as a function on the whole plane okay the zoomed limit function is on the whole plane so I have this okay now what I want you to notice is that the first thing I want to tell you is that this zoomed limit function what is each zoomed limit function it is a normal limit of meromorphic functions okay but it is a normal limit of functions from fk script f sub k but mind you script f sub k are all analytic okay see because we cleverly assumed one of the omitted values is infinity and therefore we are only working with analytic functions okay therefore this limit function these limit functions gks they also have to omit the value infinity the only other possibility is that they can be identically infinity because you know whenever you have a normal limit of analytic functions okay then either the normal limit is again and the limit function is again analytic or it is identically infinity this is the only thing that is possible so the only thing that could have happened is that these limit functions are all identically infinite some of the limit functions as you zoomed limit functions gks they could have been identically infinity but if it if but even that cannot happen because if they were identically infinity the spherical derivative would have been 0 but I have put the condition that the Zalkman Slema tells you that the spherical derivative at the origin is 1 they are non concept so what it means is that all these gks are all entire they are all entire functions you have you have cooked up the family of entire functions you have cooked up a sequence of entire functions okay so let me write that down note that since gk hash of 0 is 1 and gk is a normal limit of analytic functions from the family fk gk is entire it is entire because it is analytic and it is defined on the whole plane so it is entire and the point is that it does not take the value infinity cannot take the value infinity see in principle it could have been a meromorphic function it could have taken the value infinity at a pole but it can never take the value the only way the only possibility is that because it is a normal limit not of just meromorphic functions but it is a normal limit of analytic functions okay the limit can only either be completely analytic or it can be completely identical infinity you cannot get from a limit of normal limit of analytic functions you cannot cook up a meromorphic function this will not happen that is basically because of Hurwitz theorem because if you cook up a meromorphic function it means that your pole is popping up the limit but if a pole pops up then for the reciprocal function a 0 pops up Hurwitz theorem says that a 0 of the limit will come from the 0 of the original functions beyond a certain stage that means the original functions the reciprocal of the original functions if they if the limit takes if the limit takes value 0 or if the limit takes the value infinity okay then the reciprocal of the limit will take the value 0 and the reciprocal of the original functions beyond a certain stage should have zeros which means that beyond a certain stage the original functions should be meromorphic but they are all analytic okay so basically it is Hurwitz theorem which is working behind all this so therefore each of these functions is entire and the beautiful thing is what are the values that they miss see these gk will miss the value 0 infinity of course and all the 2k th roots of unity because every function in fk script fk is suppose by construction it misses all the 2k th roots of unity okay so let me write that down note that gk misses the value 0 and infinity 0 and the 2k th roots of unity and actually what I am using here is actually Hurwitz theorem use Hurwitz theorem so again let me repeat that what does Hurwitz theorem says say you take a sequence of analytic functions okay suppose it converges normally to a limit function okay a normal limit of analytic functions is always analytic okay or it can be identically infinity but this being identically infinity is anyway out of the picture because all the spherical derivatives are all nonzero okay so the limit is always an analytic function okay so if you have a sequence of analytic functions that converge normally to an analytic function then the limit function if it has a 0 then the 0 must come by a limit of 0s of the original functions that are converging beyond a certain stage that is Hurwitz's theorem okay in other words what are you saying here what is it saying it is saying that you know if the limit function takes the value 0 then the original function should also take the value 0 beyond a certain stage in a neighbourhood of the 0 of the limit function okay and this is not only true for the value 0 it is true for any value because the f for f of z to take the value lambda it is the same as looking at a 0 of f of z minus lambda which is also analytic okay so actually what you can say is Hurwitz's theorem can also be thought of as suppose you have a sequence of analytic functions suppose it is converging normally to a non-constant analytic function okay then if the limit function takes any complex value then all the original functions also should take that complex value beyond a certain stage that is what it says and in fact Hurwitz's theorem says more in fact it says that even the multiplicity is to coincide the multiplicity if the limit function takes a value with a certain multiplicity then all the functions in the original sequence that converge to that limit function they also should take the same value with the same multiplicity beyond a certain stage okay in a neighbourhood of the point where that value is assumed okay so this is just Hurwitz's theorem okay so mind you spherical derivative is a non-negative real valued function okay the limit function GK these are all these are the ones that are entire and they miss the values infinity 0 and the 2K roots of unity okay and of course I will have to make use of these two conditions here that the they all have spherical derivative 1 and 0 and they all have spherical derivative bounded by 1 so note that all these GKs having spherical derivative bounded by 1 what does it tell you you can apply Marty's theorem now you look at the sequence of GKs this is a sequence of an entire functions on the plane okay GK is a sequence of entire functions on the plane they are spherical derivatives are all bounded therefore by Marty's theorem they there is a subsequence which will converge normally on the plane okay so now I am applying Marty's theorem okay by Marty's theorem this GK has a convergent subsequence normally convergent subsequence G and K on of course it is normally convergent on the whole plane okay because GK is of course it is a family of entire functions okay let us take so take such a normally convergent subsequence and take a limit take the limit function okay you will get a function G alright now that function G see that function G is now a normal limit of entire functions okay therefore the only possibility is that it is also entire okay or it is identically infinity but it cannot be identically infinity because of the spherical derivative being 1 at 0 so the limit function is also going to be an entire function okay and you will see that is the function for which I am going to apply Lieuville's theorem and get constancy which is a contradiction okay so let G be limit as K tends to infinity of G and K okay then G is entire then of course you know G hash is also bounded by 1 and G hash at the origin is 1 so G is entire non-constant and non-constant okay so you have cooked up an entire non-constant function okay and see now comes the now comes something very nice see each of the G and K's the value is that they omit are 0 infinity and the 2Kth 2 to the 2 to the NKth roots of unity okay but you know as K becomes large these 2 to the NKth roots of unity they if you take the union of all these that is a dense subset of the unit circle okay so what it means is that this G this function G it omits a dense subset of values of the unit circle therefore it has to omit all values on the unit circle this is because of the open mapping theorem what does open mapping theorem say whenever a function takes a value it has to take all values in a small disc surrounding that the image of every open set is an open set for a non-constant analytic function the image of an open set is always an open set okay so G being an entire non-constant entire function if G omits values on a dense subset of the unit circle by the open mapping theorem G has to omit all the values on the unit circle okay but the unit circle disconnects the plane into 2 pieces one is the interior and the other is the exterior and therefore the image of the complex plane under G which has to be connected has to either go completely inside the unit disc or it has to go completely outside the unit disc if it goes inside the unit disc then you have found G is a bounded entire function so it is a constant that is a contradiction if it goes completely outside the unit disc you take 1 by G which is also going to be entire because mind you G also omits the value 0 so 1 by G is also entire so 1 by G will become a bounded entire function it will become constant therefore G will become a constant so in any case you get G is a constant you get a contradiction okay and that proves the theorem that is all okay so what you must appreciate is you have used open mapping theorem you have used Hurwitz's theorem you have used Salkman's lemma you have used the Marthe's theorem okay everything has been used okay so let me write this down since Gk so G to the nk omits the omits the values 0 infinity and 1 by 1 by 2 to the nk G omits the values 0 infinity and mod Z is equal to 1 because of the open mapping theorem which says G is open and the fact that 1 by the 2 power nk th roots of unity k equal to 1 2 and so on is a dense subset of mod Z is equal to 1 thus G of C which is connected and belongs to complex plane minus mod Z is equal to 1 has to imply either mod G is less than 1 or mod G is greater than 1 okay if mod G is less than 1 Liewel's theorem implies G is equal to constant contradiction it is not constant because spherical derivative at the origin is 1 okay and if G is if mod G is greater than 1 then 1 by G is entire and mod 1 by G is less than 1 so again Liewel's theorem implies 1 by G is constant which implies G is constant again contradiction okay so that is it so the family has to be normal right so that finishes the proof of this theorem and as a corollary you can see that if you have a family of analytic functions on a domain if you know that the family omits 2 complex values 2 finite complex values then it has to be normal that is a corollary of this okay this theorem that we have proved is for meromorphic functions and you are including the value infinity also okay so sometimes that is called as if this condition of omitting 3 values being omitted for a family of meromorphic functions or 2 values being omitted for a family of analytic functions is called the fundamental normality criterion so it is a condition very simple condition to check whether a given family is normal okay I will stop here.